A&A 397, 353-359 (2003)

DOI: 10.1051/0004-6361:20021518

**P. Oberti ^{1} - A. Vienne^{2}**

1 - Observatoire de la Côte d'Azur,
BP 4229, 06304 Nice Cedex 4, France

2 -
Institut de mécanique céleste et de calcul des éphémérides,
UMR 8028 du CNRS and
Université de Lille, 1 impasse de l'Observatoire, 59000 Lille, France

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

Recently published observations of Tethys and Dione's Lagrangian satellites Telesto, Calypso, and Helene (Veiga & Vieira Martins 2000; Veiga et al. 2002), give an opportunity to revisit and improve the theory of their motions to face an almost 2-decade observation time span.

The mass ratios Tethys/Saturn
(
)
and Dione/Saturn
(
)
ensure the stability of the Lagrangian points *L*_{4}and *L*_{5} in the Restricted Circular 3-Body Problem. Due to the 1:1-resonance
with the companion, a satellite around these positions shows librations in its
motion resulting in a tadpole-shaped orbit in a rotating reference frame.
Expansions around the equilibrium positions provided accurate solutions in the
bidimensional case (Deprit & Rom 1970).

Pointed out as a significant perturbation by numerical simulations, Saturn's oblateness (through the first zonal coefficient of Saturn's potential developed in spherical harmonics) was then introduced in the model by addition of perturbing terms in the second and following orders of the previously expanded non-perturbed Hamiltonian (Oberti 1990).

This new theory, now developed in a tridimensional context, deals with
the perturbation at the uppermost level of the expansion, the quadratic part of
the Hamiltonian (generally called *H*_{0}), improving the convergence of the
solution and offering an explicit generalization suitable for other
studies.

Moreover, the model could prove useful for the analysis of the CASSINI space probe's accurate observations of these faint satellites that should be obtained during its nearing exploration of the Saturnian system.

For the sake of convenience, computations are carried out in
dimensionless units (Szebehely 1967). The distance between the
primaries, and the sum of their masses (called
for the second
primary and
for Saturn), are assumed to be unity. The reference
frame is a barycentric rotating frame with a rotational period of
(in
dimensionless time ), where the value of *k* will be computed in order to
insure a consistent model involving fixed primaries. The *xy*-plane contains the
primaries, the *x*-axis being located along the primaries and positively
oriented from Saturn toward the second primary.

The theory will be expanded around *L*_{4}. Dealing with the symmetrical
point *L*_{5} requires only slight modifications indicated in Sect. 7.

With
,
*L*_{4} is located at the point
.

Let *x*, *y*, *z* be the satellite's Cartesian coordinates and *P*_{x}, *P*_{y}, *P*_{z} their conjugate variables in the barycentric rotating frame.
The Hamiltonian of the motion is:

where:

and stands for half the first Saturnian zonal harmonic coefficient

*L*_{4} is no longer an equilibrium position (Sharma & Subba Rao
1976). The conditions:

lead to:

where dots stand for derivation with respect to the dimensionless time. Canceling the accelerations would require two different values of

The value of

where:

when:

With

A first canonical transformation defines a coordinate system in the vicinity of

where

Applying the transformation turns the Hamiltonian into:

where:

Using, for example, the software *Mathematica*, we expand the
functions of the coordinates in the vicinity of
:

and similar expressions for with the second index. Then, scaling with respect to the total power

Neglecting the remainder:

Defining:

the substitutions into

The terms in and , resulting from the fact that

Removing the -term leads to:

and

where:

Then, let us focus on the linear system deriving from the quadratic Hamiltonian

leading to imaginary solutions . The zero-order frequencies of the motion are:

and

by a canonical transformation of the form:

The required expression for

where:

Finally, the canonical transformation:

leads to:

The zero-order Hamiltonian

Normalizing *H*_{0}, the
-variables are suitable for
performing a *n*th-order Lie transformation
on the approximation *H*_{(n)} of the Hamiltonian of the motion (Deprit 1969). First, the last two
canonical transformations are applied to the components *H*_{1}, *H*_{2}, ...,
*H*_{n} for dealing with
-variables. Then, the Lie algorithm is
carried out using an algebraic processor from the University of Namur (Moons
1991). It results in a *n*th-order normalized Hamiltonian depending
only on the variables
,
providing *n*th-order solutions in the
vicinity of the equilibrium position.

The Lie algorithm was carried out up to the fourth order mainly to accurately compute the frequencies of the motion (they are only updated at even steps). It gives the theory a better reliability for prediction. Non-significant solution terms (with respect to the statistics of the solution) will be removed after the fit to the observations.

The development process is a bit less straightforward than in the case
of the geosynchronous satellite because of extra terms that are not canceled by
the orientation of the *normal* frame (Oberti 1994). To increase
the convergence, each even generator has to be completed by polynomials in the
action variables devoted to removing solution terms having the unfortunate
propensity to grow too fast.

The value of the theory in the dimensionless space, at a selected time
,
depends on six *internal* parameters: three amplitudes *I*^{0}_{1},
*I*^{0}_{2}, *I*^{0}_{3}, and three phases
,
,
(dropping the *primes* to save writing). When going back to the real world,
another two (*external*) parameters can be fitted for fine tuning: the
companion's longitude
at
(its initial value is provided by
the companion's theory) and the scaling factor *r*_{0} between the theoretical
space and the actual one (its initial value is computed from the companion's
observed period).

With an expansion around the equilibrium position, the accuracy of the
theory strongly depends on the satellite's libration amplitude, and its ability
to correctly represent the motion increases with the development order. To check
the *inner* limits of the theory (due to the finite expansion of the
Hamiltonian), the solution was fitted by least squares (*internal*
parameters only) to a numerical integration of the same Hamiltonian system over
twenty years.

With initial values causing libration amplitudes of 10, 20, 30, 40, and 50
degrees, the accuracy is of order 10^{-6}, 10^{-4}, 10^{-3}, 10^{-2},
and
arcsec. Helene's libration amplitude being of
order 30 degrees, and Telesto and Calypso's much smaller, the
theory qualifies for the job regarding this test.

The theory assumes a non-Keplerian circular motion for the second
primary and doesn't take into account the neighboring satellites. To check the
*outer* limits of the theory (due to this more complex environment), the
solution was fitted by least squares to simulated observations generated at each
observation time by a numerical integration of the studied satellite. The
numerical model, including the first six Saturn's satellites (Vienne
& Duriez 1995; Duriez & Vienne 1997) was previously fitted
to the actual observations.

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 value is of order 0.15 arcsec for
the three satellites. Allowing the fit of the companion's initial longitude
removes this dependency, and the precision is about 0.03 arcsec for
Helene, 0.05 arcsec for Telesto, and 0.08 arcsec for
Calypso. The effect 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.

Here again, the theory pass the test for the three satellites.

The last test shows the relevance of the model: without or with
Saturn's oblateness (*I*_{J}=0 or 1) and bidimensional or tridimensional
(*I*_{D}=2 or 3). To remove any side effects, the companion's longitude and the
scaling factor were fitted in every case. The resulting precisions are
summarized in Table 1, in arcsec, for the different values of
*I*_{J}/*I*_{D}. Comparatively to the *basic* model (0/2), only the combination
of the two extensions seems to make the difference...

Model | Helene | Telesto | Calypso |

0/2 | 0.20 | 0.60 | 0.75 |

0/3 | 0.15 | 0.40 | 0.65 |

1/2 | 0.10 | 0.55 | 0.60 |

1/3 |
0.03 |
0.05 |
0.08 |

They are performed using the observations from Reitsema (1981a, 1981b) obtained in 1980 and 1981 for Helene (20 obs.), Telesto (2 obs.), and Calypso (4 obs.), the observations from Oberti et al. (1989) obtained in 1981, 1982, 1984, and 1985 for Helene (133 obs.), Telesto (46 obs.), and Calypso (93 obs.), the observations from Veiga & Vieira Martins (2000) obtained in 1985 and 1987 for Helene (22 obs.), and the observations from Veiga et al. (2002) obtained in 1995 and 1996 for Helene (46 obs.), Telesto (13 obs.), and Calypso (19 obs.), the latter being deduced from the positions of the major satellites as exposed in Vienne et al. (2001).

For each satellite, three different fits were performed using first the observations from 1980 to 1987, then the observations from 1995 to 1996, and finally all the observations. The precisions of the resulting solutions are summarized in Table 2, in arcsec. The residuals from Veiga et al. (2002) were 0.15 arcsec for Helene, 1.43 arcsec for Telesto, and 0.42 arcsec for Calypso.

The selected solutions are the ones computed from the whole set of observations. Deduced from these solutions, the respective libration angles for Helene, Telesto, and Calypso are of order 28.5, 4.5, and 10.5 degrees, and their respective long-period libration periods are of order 783, 662.5, and 663.5 days.

Data | Helene | Telesto | Calypso |

1980-1987 | |||

1995-1996 | |||

1980-1996 |

For each satellite, the fitted solution gives the Cartesian
coordinates and their conjugate variables in the theoretical space. They are
developed in trigonometric polynomials involving three time-depending angles
,
,
:

where is the dimensionless time computed from the

with

Obtaining the saturnicentric positions *X*, *Y*, *Z* and velocities
,
,
(where dots stand for derivation with respect to
the real time) referred to the mean ecliptic and equinox of J2000 requires
several steps. With the notations of the first canonical transformation:

where

and

The signs of *y* and *P*_{x} are changed for Calypso to take into account
its position near *L*_{5}.

Then, the series are shortened giving the three amplitudes *I*^{0}_{1},
*I*^{0}_{2}, *I*^{0}_{3} their numerical values and removing non-significant terms, and
the frequencies of the four angles are multiplied by
(and called
,
,
,
and )
to deal directly with a more realistic
time.

The signs of ,
,
and
are changed for Calypso to
take into account its position near *L*_{5}.

The frequencies (rd/day) and phases (rd) for Helene, Telesto, and Calypso, together with their saturnicentric positions (a.u.) and velocities (a.u./day), referred to the mean ecliptic and equinox of J2000, are summarized in Tables 3-8.

Each coordinate is the sum of the corresponding numerical coefficients
multiplied by the cosine (*f*=0) or sine (*f*=1) of the angle
computed
by:

where

A downloadable FORTRAN code for the theory is available at: ftp://www.imcce.fr/pub/ephem/satel/htc20/

The comparison with numerical simulations shows the model's efficiency to deal with Lagrangian satellites similar to Tethys and Dione's companions (with small and moderate libration amplitudes). With almost twenty years of observations, the precision obtained when fitting each satellite's theory gives a rather good level of confidence for predictions.

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- Deprit, A. 1969, Celest. Mech., 1, 12 In the text NASA ADS
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- Duriez, L., & Vienne, A. 1997, A&A, 324, 366 In the text NASA ADS
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- Vienne, A., Thuillot, W., Veiga, C. H., Arlot, J.-E., & Vieira Martins, R. 2001, A&A, 380, 727 In the text NASA ADS
- Szebehely, V. 1967, Theory of orbits (Academic Press) In the text

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