An Upgraded Theory for Helene, Telesto, and Calypso

Home , Calypso (moon), Helene (moon), Telesto (moon)

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 Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France e-mail: [email protected] 2 Institut de mécanique céleste et de calcul deseph´ ´ emérides, UMR 8028 du CNRS and Université 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. c = 1 9J − − − 2 · ! The zero-order Hamiltonian H0 represents a triple harmonic oscillator under its normalized form, providing zero-order so- Then, let us focus on the linear system deriving from the lutions in the vicinity of the equilibrium position. The resulting quadratic Hamiltonian H0. The eigenvalues are computed motion is composed of two elliptic motions in the ξη-plane, solving: one with a short period (2π/ω1) and one with a long period (2π/ω ), completed by an oscillation along the ζ-axis with a λ4 + 2k2 a b λ2 + ab + k2a + k2b + k4 = 0, 2 − − short period (2π/ω3). λ2 c = 0, − 4. Asymptotic expansions leading to imaginary solutions i ω1, i ω2, i ω3. The zero- ± ± ± Normalizing H0,the(I,ϕ)-variables are suitable for performing order frequencies of the motion are: a nth-order Lie transformation (I,ϕ) (I ,ϕ) on the approx- → 0 0 imation H(n) of the Hamiltonian of the motion (Deprit 1969). 2k2 a b + d ω = − − 1, First, the last two canonical transformations are applied to the 1 2 ∼ r components H1, H2, ..., Hn for dealing with (I,ϕ)-variables. Then, the Lie algorithm is carried out using an algebraic pro- 2 2k a b d cessor from the University of Namur (Moons 1991). It results ω2 = − − − √µ, r 2 ∼ in a nth-order normalized Hamiltonian depending only on the variables (I0), providing nth-order solutions in the vicinity of ω = √ c 1, d = (a b)2 8k2 (a + b), the equilibrium position. 3 − ∼ − − q The Lie algorithm was carried out up to the fourth order and H0 can be changed into: mainly to accurately compute the frequencies of the motion (they are only updated at even steps). It gives the theory a better 1 2 2 2 1 2 2 2 1 2 2 2 reliability for prediction. Non-signiﬁcant solution terms (with H0 = P1 + ω1Q1 P2 + ω2Q2 + P3 + ω3Q3 2 − 2 2 respect to the statistics of the solution) will be removed after by a canonical transformation of the form: the ﬁt to the observations. The development process is a bit less straightforward than ξ = a13P1 + a14P2, Pξ = a31Q1 + a32Q2, in the case of the geosynchronous satellite because of extra terms that are not canceled by the orientation of the normal η = a21Q1 + a22Q2, Pη = a43P1 + a44P2, frame (Oberti 1994). To increase the convergence, each even generator has to be completed by polynomials in the action ζ = Q3, Pζ = P3. variables devoted to removing solution terms having the un- fortunate propensity to grow too fast. 356 P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso

5. Some checks Table 1. Relevance of the model IJ /ID (arcsec). The value of the theory in the dimensionless space, at a selected time τ, depends on six internal parameters: three amplitudes Model Helene Telesto Calypso 0 0 0 0 0 0 I1 , I2 , I3 , and three phases ϕ1, ϕ2, ϕ3 (dropping the primes to 0/20.20 0.60 0.75 save writing). When going back to the real world, another two 0/30.15 0.40 0.65 (external) parameters can be fitted for fine tuning: the compan- 1/20.10 0.55 0.60 ion’s longitude θ0 at τ = 0 (its initial value is provided by the 1/30.03 0.05 0.08 companion’s theory) and the scaling factor r0 between the theoretical space and the actual one (its initial value is computed from the companion’s observed period). Telesto (2 obs.), and Calypso (4 obs.), the observations from With an expansion around the equilibrium position, the ac- Oberti et al. (1989) obtained in 1981, 1982, 1984, and 1985 curacy of the theory strongly depends on the satellite’s libration for Helene (133 obs.), Telesto (46 obs.), and Calypso (93 obs.), amplitude, and its ability to correctly represent the motion in- the observations from Veiga & Vieira Martins (2000) obtained creases with the development order. To check the inner limits in 1985 and 1987 for Helene (22 obs.), and the observations of the theory (due to the finite expansion of the Hamiltonian), from Veiga et al. (2002) obtained in 1995 and 1996 for Helene the solution was fitted by least squares (internal parameters (46 obs.), Telesto (13 obs.), and Calypso (19 obs.), the latter only) to a numerical integration of the same Hamiltonian sys- being deduced from the positions of the major satellites as ex- tem over twenty years. posed in Vienne et al. (2001). With initial values causing libration amplitudes of 10, 20, For each satellite, three different fits were performed using 6 4 30, 40, and 50 degrees, the accuracy is of order 10− ,10− , first the observations from 1980 to 1987, then the observations 10 3,10 2,and5 10 2 arcsec. Helene’s libration amplitude from 1995 to 1996, and finally all the observations. The preci- − − × − being of order 30 degrees, and Telesto and Calypso’s much sions of the resulting solutions are summarized in Table 2, in smaller, the theory qualifies for the job regarding this test. arcsec. The residuals from Veiga et al. (2002) were 0.15 arcsec The theory assumes a non-Keplerian circular motion for the for Helene, 1.43 arcsec for Telesto, and 0.42 arcsec for second primary and doesn’t take into account the neighboring Calypso. satellites. To check the outer limits of the theory (due to this The selected solutions are the ones computed from the more complex environment), the solution was fitted by least whole set of observations. Deduced from these solutions, the squares to simulated observations generated at each observa- respective libration angles for Helene, Telesto, and Calypso are tion time by a numerical integration of the studied satellite. of order 28.5, 4.5, and 10.5 degrees, and their respective long- The numerical model, including the first six Saturn’s satellites period libration periods are of order 783, 662.5, and 663.5 days. (Vienne & Duriez 1995; Duriez & Vienne 1997) was previously fitted to the actual observations. Table 2. Results of the fits to the observations (arcsec). When only the six internal parameters are fitted, the precision strongly depends on the initial time chosen for the series through the induced companion’s initial position. Its average Data Helene Telesto Calypso value is of order 0.15 arcsec for the three satellites. Allowing 1980–1987 0.27 0.27 0.28 0.26 0.25 0.22 ± ± ± the fit of the companion’s initial longitude removes this de- 1995–1996 0.13 0.19 0.40 0.45 0.25 0.29 pendency, and the precision is about 0.03 arcsec for Helene, 1980–1996 0.65 ± 0.79 0.36 ± 0.35 0.83 ± 1.03 0.05 arcsec for Telesto, and 0.08 arcsec for Calypso. The ef- ± ± ± fect of the last parameter (the scaling factor) is barely visible. Despite a greater libration amplitude, Helene’s motion seems to be a bit more accurately modeled, possibly because the motion of Tethys is less regular than Dione’s. 7. Back to physical variables Here again, the theory pass the test for the three satellites. For each satellite, the fitted solution gives the Cartesian coor- The last test shows the relevance of the model: without or dinates and their conjugate variables in the theoretical space. with Saturn’s oblateness (IJ = 0 or 1) and bidimensional or They are developed in trigonometric polynomials involving tridimensional (ID = 2 or 3). To remove any side effects, the three time-depending angles ϕ1, ϕ2, ϕ3: companion’s longitude and the scaling factor were fitted in ev- 0 0 0 ery case. The resulting precisions are summarized in Table 1, ϕ1 = ω1τ + ϕ1,ϕ2 = ω2τ + ϕ2,ϕ3 = ω3τ + ϕ3, in arcsec, for the different values of IJ/ID. Comparatively to the where τ is the dimensionless time computed from the Saturnian basic model (0/2), only the combination of the two extensions Julian day t (Terrestrial Time minus light time) by: seems to make the difference. . . 2π t t τ = − 0 , k P 6. Fits to the observations with t0 = 2 451 545.0, P being the companion’s observed pe- They are performed using the observations from Reitsema riod (in days): 1.887803 for Tethys and 2.736916 for Dione (1981a, 1981b) obtained in 1980 and 1981 for Helene (20 obs.), (Vienne & Duriez 1995). P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso 357

Table 3. Helene’s frequencies (rd/day) and phases (rd).

0 0 0 ν1 ν2 ν3 λϕ1 ϕ2 ϕ3 θ0 2.29427177 0.00802443 2.29714724 2.29571726 3.27342548 1.30770422 0.77232982 3.07410251 −

Table 4. Helene’s positions (a.u.) and velocities (a.u./day).

fX Y Z X˙ Y˙ Zn˙ 1 n2 n3 ` 0 0.002396 0.000399 0.000442 0.001278 0.004939 0.002466 0 0 0 1 − − − 10.000557 0.002152 0.001074 0.005500 0.000916 0.001015 0 0 0 1 1 0.000003− 0.000003 0.000011− 0.000006 1 0 0 1 0 −0.000066 0.000265 0.000133 0.000676 −0.000107 0.000122 0 1 0 1 − − − − 1 0.000295 0.000047 0.000053 0.000151 0.000607 0.000303 0 1 0 1 00− .000015− 0.000017 0.000010 0.000044− 0.000033 0.000013 0 2 0 1 1 0.000019 0.000014 −0.000006 −0.000035 0.000038− 0.000023 0 2 0 1 − − − − 00.000002 0.000002 0.000004 0.000002 0 3 0 1 0 0.000002 0.000008 0.000004− − 1 0 0 1 10− .000009 −0.000002 1 0 0 −1 − − 0 0.000067 0.000264 0.000132 0.000677 0.000110 0.000123 0 1 0 1 10− .000294 0.000048 −0.000053 −0.000154− 0.000608 0.000304 0 1 0 −1 00.000015 0.000016 −0.000010 −0.000044 0.000033 −0.000013 0 2 0 −1 − − − − 10.000019 0.000014 0.000006 0.000035 0.000038 0.000023 0 2 0 1 00.000002 − 0.000002 0.000004 −0.000002 0 3 0 −1 10.000005 0.000010− − 0 0 1− 0 00.000002 0.000013 0.000002 0.000002 1 0 0 1 10.000002 −0.000004 −0.000002 0 3 0 1 1 0.000002− 0.000004− 0.000002 0 3 0 1 − − 1 0.000003 1 1 0 1 1 −0.000003 1 10 1 00− .000005 0.000010 0− 0 1 0 00.000003 1 1 0 1 00.000003 1 10 1 −

Obtaining the saturnicentric positions X, Y, Z and velocities Then, the series are shortened giving the three amplitudes ˙ ˙ ˙ 0 0 0 X, Y, Z (where dots stand for derivation with respect to the I1 , I2 , I3 their numerical values and removing non-significant real time) referred to the mean ecliptic and equinox of J2000 terms, and the frequencies of the four angles are multiplied by requires several steps. With the notations of the first canonical 2π/kP (and called ν1, ν2, ν3,andλ) to deal directly with a more transformation: realistic time. 1 The signs of ν1, ν2,andν3 are changed for Calypso to take X x + 2 (1 γ) − into account its position near L . Y = R R r y , 5   2 1  0    Z    z            8. User’s manual  X˙    Px  ˙ 2π 1 Y = R2 R1 r0 Py + 2 (1 γ) , The frequencies (rd/day) and phases (rd) for Helene, Telesto,    kP  −  and Calypso, together with their saturnicentric positions (a.u.)  Z˙    Pz       and velocities (a.u./day), referred to the mean ecliptic and      where  r0 is the scaling factor, R1 is a time-depending rotation equinox of J2000, are summarized in Tables 3–8. of angle θ in the ring plane devoted to returning to the fixed Each coordinate is the sum of the corresponding numerical frame (and adding a new trigonometric variable in the series), coefficients multiplied by the cosine ( f = 0) or sine ( f = 1) of where: the angle ϕ computed by:

θ = kτ + θ0, 0 0 ϕ = n1(ν1T + ϕ1) + n2(ν2T + ϕ2) and R is a composition of numerical rotations linking the ring 0 2 +n3(ν3T + ϕ3) + `(λT + θ0), plane to the chosen frame. The signs of y and Px are changed for Calypso to take into where T is a Saturnian time (Terrestrial Time minus light time) account its position near L5. in days computed from 2 451 545.0. 358 P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso

Table 5. Telesto’s frequencies (rd/day) and phases (rd).

0 0 0 ν1 ν2 ν3 λϕ1 ϕ2 ϕ3 θ0 3.32489098 0.00948045 3.33170385 3.32830561 6.24233590 4.62624497 0.04769409 3.24465053 −

Table 6. Telesto’s positions (a.u.) and velocities (a.u./day).

fX Y Z X˙ Y˙ Zn˙ 1 n2 n3 ` 10.000002 0.000010 0.000019 0 0 1 0 0 0.001933 0.000253 0.000320 0.001237 0.005767 0.002904 0 0 0 1 10− .000372 −0.001733 0.000873 0.006432− 0.000842 0.001066 0 0 0 1 1 0.000002− 0.000003 0.000014− 0.000007 1 0 0 1 − − 0 0.000006 0.000029 0.000015 0.000108 0.000014 0.000018 0 1 0 1 1 −0.000033 0.000004− 0.000005− 0.000020 −0.000097 0.000049 0 1 0 1 10− .000007− − 1 0 0 1 − 0 0.000006 0.000029 0.000015 0.000108 0.000014 0.000018 0 1 0 1 10− .000032 0.000004 −0.000005 −0.000021− 0.000097 0.000049 0 1 0 −1 00.000002 − −0.000016 0.000002− 0.000003 1 0 0− 1 − − 00.000007 0.000003 1 0 0 1 00− .000002 0.000010 0.000019 0 0 1− 0

Table 7. Calypso’s frequencies (rd/day) and phases (rd).

0 0 0 ν1 ν2 ν3 λϕ1 ϕ2 ϕ3 θ0 3.32489617 0.00946761 3.33170262 3.32830561 5.41384760 1.36874776 5.64157287 3.25074880 − −

Table 8. Calypso’s positions (a.u.) and velocities (a.u./day).

fX Y Z X˙ Y˙ Zn˙ 1 n2 n3 ` 10.000005 0.000027 0.000052 0 0 1 0 00.000651 0.001615 0.000910 0.006145 0.002170 0.000542 0 0 0 1 0 0.000011 0.000004− − − 1 0 0 1 1 −0.001846 0.000652 0.000163 0.002166 0.005375 0.003030 0 0 0 1 − − − − 1 0.000004 0.000010 0.000006 1 0 0 1 0 −0.000077− 0.000028 0.000007 0.000092 0.000225 0.000127 0 1 0 1 1 −0.000028 0.000067− 0.000038− 0.000257 −0.000092 0.000023 0 1 0 1 − − − 0 0.000002 0.000004 0.000006 0.000003 0 2 0 1 0 −0.000004 0.000009 −0.000022 0.000012 1 0 0 1 0 −0.000078 0.000027 0.000007 −0.000089 −0.000225 0.000127 0 1 0 −1 − − − − − 10.000027 0.000068 0.000038 0.000257 0.000089 0.000022 0 1 0 1 0 0.000002− − 0.000004 0.000006− 0.000003 0 2 0 −1 1 − 0.000002 0.000007− 0.000003 0.000002 0 2 0− 1 − − 10.000003 0.000002 0.000025 0.000009 0.000002 1 0 0 1 10.000002 − −0.000007 0.000003− 0.000002 0 2 0 −1 0 0.000002− − 0 1 1− 0 − 0 0.000002 0 1 10 00− .000005 0.000027 0.000052 0 0− 1 0 00.000002 1 1 0 1 − 00.000002 1 101 1 0.000002 1− 1 0 −1 1 −0.000002 1 10−1 − − − 10.000002 0 1 1 0 1 0.000002 0 1 10 − − P. Oberti and A. Vienne: An upgraded theory for Helene, Telesto, and Calypso 359

A downloadable FORTRAN code for the theory is available Deprit, A., & Rom, A. 1970, A&A, 5, 416 at: ftp://www.imcce.fr/pub/ephem/satel/htc20/ Duriez, L., & Vienne, A. 1997, A&A, 324, 366 Moons, M. 1991, Dept. of Math., Facultés Universitaires Notre Dame de la Paix, Namur, Belgium 9. Conclusion Oberti, P., Veillet, C., & Catullo, V. 1989, A&AS, 80, 289 Oberti, P. 1990, A&A, 228, 275 The comparison with numerical simulations shows the model’s Oberti, P. 1994, A&A, 284, 281 efficiency to deal with Lagrangian satellites similar to Tethys Reitsema, H. J. 1981a, Icarus, 48, 23 and Dione’s companions (with small and moderate libration Reitsema, H. J. 1981b, Icarus, 48, 140 amplitudes). With almost twenty years of observations, the Sharma, R. K., & Subba Rao, P. V. 1976, Celest. Mech., 13, 137 precision obtained when fitting each satellite’s theory gives a Veiga, C. H., & Vieira Martins, R. 2000, A&AS, 143, 405 rather good level of confidence for predictions. Veiga, C. H., Vieira Martins, R., Vienne, A., Thuillot, W., & Arlot, J.-E. 2002, preprint Vienne, A., & Duriez, L. 1995, A&A, 297, 588 References Vienne, A., Thuillot, W., Veiga, C. H., Arlot, J.-E., & Vieira Martins, Campbell, J. K., & Anderson, J. D. 1989, AJ, 97, 1485 R. 2001, A&A, 380, 727 Deprit, A. 1969, Celest. Mech., 1, 12 Szebehely, V. 1967, Theory of orbits (Academic Press)