A&A 379, 245-256 (2001)
DOI: 10.1051/0004-6361:20011336
P. Morel - G. Berthomieu - J. Provost - F. Thévenin
Département Cassini, UMR CNRS 6529, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France
Received 9 July 2001 / Accepted 14 September 2001
Abstract
We have revisited the calibration of the
visual binary system Herculis with the goal to
give the seismological properties of
the G0IV sub-giant
HerA. The sum of masses and the mass fraction
are derived from the most recent astrometric data mostly based on the
HIPPARCOS ones. We have derived the effective temperatures,
the luminosities and the metallicities
from available spectroscopic data and TYCHO
photometric data and calibrations. For the calculations of
evolutionary models we have used updated physics and the most recent physical
data. A
minimization is performed to approach the most reliable
modeling parameters which reproduce the observations
within their error bars. For the age of the
Her binary system we have
obtained
Myr, for the masses
and
,
for the initial helium mass fraction
,
for the initial mass
ratio of heavy elements to hydrogen
and for the
mixing-length parameters
and
using the Canuto &
Mazitelli (1991, 1992) convection theory.
Our results do not exclude that
HerA is itself
a binary sub-system as has been suspected many times in the past century;
the mass of the hypothetical unseen companion would be
,
a value significantly smaller
than previous determinations.
A calibration made with an overshoot of the convective core of
HerA leads to similar results but with a slight increase of
+250 Myr for the age.
The adiabatic oscillation spectrum of
HerA is found to be a
complicated superposition of
acoustic and gravity modes. Some of these waves have a dual character.
This greatly complicates the classification of the non-radial modes.
For
the modes all have both
energy in the core and in the envelope; they are mixed modes.
For
there is a succession of modes with energy either in the core or in the
envelope with a few mixed modes.
The echelle diagram used by the observers to extract the frequencies
will work for
.
The large difference
is found to be of the order of
Hz, in
agreement with the Martic et al. (2001) seismic observations.
Key words:
stars: binaries: visual - stars: evolution -
stars: fundamental parameters - stars: individual: Her
Herculis (40Her; BD+312884; STF2084; ADS10157;
IDS16375+3147; WDS 16413+3136; HR6212; HD150680; HIP81693;
,
(2000))
is a well known bright visual and single lined
spectroscopic binary system of naked-eye brightness.
According to the CDS Simbad data base
the system is composed of a 2.90 Vmagnitude G0IV sub-giant star and by a 5.53
Vmagnitude G7V dwarf star.
The binarity was discovered by
Herschell as early as 1782 (Aitken 1932) and the system has been carefully observed
by visual binary observers for more than six revolutions
back to the first reliable measurements by
W. Struve in 1826. Several orbital solutions
have been published from the early nineties to the present day.
The latest are by Heintz (1994) and
Söderhjelm (2000). The orbital elements are well determined.
They have recently allowed improvements of
Sproul Observatory and HIPPARCOS trigonometrical parallaxes.
About a century ago, a duplicity of
HerA was detected
from micrometer and meridian observations (Lewis 1906).
A period
yr and a semi-major axis of
was obtained for the sub-binary system
HerAa. Ten years later,
Comstock (1917) noted that the small irregularities of the areal velocity
in the orbit have the effect of an invisible companion having a period
of 18yr and an amplitude less than
.
Later,
a careful reanalysis of the available
astrometric and spectroscopic observational material led Berman (1941) to
conclude that"[
] the presence of a third body revolving about the brighter
component of
Herculis is not definitely indicated''. The comparison
between the observed position angles and distances and their values derived
from its astrometric orbit allowed Baize (1976) to claim that
HerA is split in two stars of respectively
and
with an orbital period
yr and a
semi-major axis
.
McCarthy (1983) reported
that
Her contains at least one unseen companion easily detected at
m by infrared speckle interferometry.
A note in the Fifth Catalog of Orbits of Visual Binary Stars
(Hartkopf et al. 2000) stipulates:
"No evidence in the speckle or Hipparcos data for the large-amplitude
third-body orbit given by Baize''.
Lebreton et al. (1993) tried to
determine both age and chemical composition of the system, modeling the
two components simultaneously, but they did not succeeded in modeling the secondary
consistently. Based on a precise spectroscopic analysis
Chmielewski et al. (1995, hereafter C95)
succeeded in modeling both components. They
derived for the age
Gyr and for the masses of
components respectively
and
.
Since 1995 the HIPPARCOS's
parallax of
Her has been available; it was recently improved by
Söderhjelm (2000).
The TYCHO & HIPPARCOS magnitudes are also available
(Fabricius & Makarov 2000); they
have been connected to the B, V and I magnitudes (Bessell 2000).
New improved theoretical data are also available, viz. opacities,
nuclear reaction rates and equation of state. The seismic observations of
HerA recently carried out by Martic et al. (2001)
indicate the presence of solar like oscillations.
The aim of the present paper is firstly to
revisit the calibration of the Her binary system
using updated physical data and theories and improved astrometrical
and photometrical observational material, and secondly to give
seismological properties of
HerA which will be useful
to exploit future asteroseismological observations.
Based on the reasonable hypothesis of a common origin for both components,
i.e. same initial chemical composition and age, the calibration of a binary
system consists of determining a consistent evolutionary history for the
double star, given (1) the positions of the two components in the HR diagram,
(2) the stellar masses and, if possible, (3) the present-day surface chemical
compositions. The goal is to compute evolutionary models that reproduce the
observations.
The calibration yields estimates for the age, the
initial helium mass fraction and the initial metallicity
which are fundamental quantities for our
understanding of the galactic chemical evolution. Within the error bars
provided by astrometry, one also determines mass values
consistent with the stellar structure modeling.
According to the convection theory applied, one derives
values for the "mixing-length parameter'' or "convection parameter''
,
ratio of the mixing-length to the pressure scale height.
Once the physics is fixed, the modeling of the two components A and B of a binary
system requires a set
of seven so-called modeling parameters:
P | a | i | e | References |
34.45yr |
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0.464 | Heintz (1994) |
34.45yr |
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0.46 | Söderhjelm (2000) |
The knowledge of individual masses of each companion is one
cornerstone of any calibration of a binary system. As illustrated by
calibrations of Cen, (e.g. Morel et al. 2000),
the resulting age is very sensitive to the values adopted
for the masses of the components. We do not reproduce here
the large bibliography on relevant astrometric data (e.g. C95); nevertheless
we emphasize the discussion concerning the estimate of masses.
For
Her we are in the fortunate position of having two
precise and independent determinations of the
trigonometrical parallax and also improved orbital elements; from these
data, two estimates of the sum of masses can be derived independently.
A first determination is provided by a standard photographic
parallax (Heintz 1994) using
an improved relative orbit and 152 mid-nights of several decades
of Sproul Observatory long focus photographic observations. The
second one is the recently improved adjustment of parallax and orbital
elements by Söderhjelm (2000)
based on HIPPARCOS data collected during 3.25yr, old ground based observations and recent
speckle-interferometry measurements.
Table 1 lists the relevant orbital elements of these two
recent astrometric orbits; they are so close
that we can safely adopt the means listed in Table 3.
The sum of masses is provided by Kepler's third law:
The outcome of standard long focus photographic parallax
measurements is the
so-called relative parallax
.
It must be reduced
to
,
the absolute parallax,
by adding a correction representing the dependence weighted parallax of
reference stars. The correction, of order 1 to 5 mas, is not accurately known
(van deKamp 1967).
For
Her the correction is
(Lippincott 1981).
With
(Heintz 1994)
that leads to a photographic absolute parallax of
(standard deviation).
The improved absolute parallax based on HIPPARCOS data and orbital
ground-based measurements amounts to
(Söderhjelm 2000).
We adopt respectively for the parallax and the distance modulus of
Her:
Her is one of the rare binary systems for which the mass fraction:
The second estimate of the mass fraction is provided by the spectroscopic orbit.
For a single lined spectroscopic binary, with known orbital elements and
parallax, the mass fraction is given by
(e.g. Heintz 1971; Scarfe et al. 1983):
The mass fraction is not derived in the analysis of Söderhjelm (2000)
despite an angular distance
larger than the grid step
(
)
of HIPPARCOS (Martin et al. 1997),
and changes of
and
respectively
in position angle and separation along the flight of the satellite.
For further investigations we shall adopt as the mass fraction of
Her the weighted mean:
With the values retained for
and
the individual masses are:
With classical 1.5m to 2.0m telescopes Her appears as a single star
under average seeing conditions because of the large magnitude
difference and of the small angular distance. Therefore
isolated spectra of each component cannot be obtained.
The available spectroscopic data in C95
allows the derivation of effective temperature only for
HerA. We start with an estimate of
from
Magain's (1987) calibration, assuming a metallicity of
and color index
(B-V)=0.65 (see Table 2).
Then we perform a standard LTE detailed analysis using the
curve of growth technique with models atmosphere from Gustafsson
et al. (1975). Equivalent widths used are from C95 and
the oscillator strengths are from Thévenin (1990).
The microturbulence is fixed to 1.5kms-1.
With the updated solar iron abundance
(Holweger 1979; Asplund 2000) we
derived the same metallicity
dex as C95. An ionization
equilibrium is obtained from curves of growth of FeI and FeII which permits us to
derive the surface gravity
.
We correct the [Fe/H] value in
Magain's (1987) formula and derived an improved temperature
value of
K and re-iterate the curve of growth
analysis.
Finally we deduce
and
.
With these improved input parameters
the scattering of the curve of growth decreases and is satisfactory.
C95 have derived a slightly smaller value for
the gravity
invoking non- LTE effects.
Such stars with solar abundances are not suspected to suffer from NLTE
overionization (Thévenin & Idiart 1999), therefore the surface
gravity of
HerA can be considered as being well determined.
It is remarkable that this new effective temperature value we derived,
using both TYCHO's photometry (Fabricius & Makarov 2000)
and spectroscopic analysis, is very close to
K, the value
obtained by C95 using both the
photometry available in 1995 and the profile of
H
corrected by the presence of the light of the secondary.
The uncertainty on
has been estimated by varying
and
in their extremes and with the accuracy of the fit resulting from
the quality of equivalent widths.
The precision obtained for the metallicity is compatible with a value of
0.05dex, which is the more pessimistic estimate in C95.
The precision obtained for the effective temperature results from
the partial derivative of Magain's (1987) temperature
formula combined with the abundance uncertainty and errors of the
TYCHO magnitudes listed Table 2. For the effective
temperature of
HerA, C95 have obtained
K,
a similar estimate.
The projected rotational velocity
of
HerA amounts
to 3.9kms-1 (Fekel 1997). Assuming a parallel axis for
rotation and orbital motion,
,
HerA is therefore a slow rotator. We can safely infer that
it is the same for the less massive B component. So we
can neglect the small rotational velocity in modeling the internal
structure of both components.
Table 2 lists the bolometric corrections, the
bolometric magnitudes and luminosities computed
with
as the solar bolometric magnitude to be used with
the Bessell et al. (1998) calibration.
The Bessell (2000) photometric calibration allows us to derive
the standard Johnson B and V magnitudes
either from
TYCHO magnitudes, or from
and
HIPPARCOS magnitudes.
The values obtained from each differ from each other by more than is
expected from the accuracy of measurements. Moreover, fixing the color index
(B-V), the effective temperatures calculated with different
photometric calibrations (Magain 1987;
Alonso et al. 1996; Flower 1996) differ by more than 150K.
This indicates that the errors on data derived with the whole procedure
may not have a standard Gaussian distribution.
Therefore the uncertainties of quantities derived from photometric data
are added in absolute value (
norm) instead of standard quadrature
(
norm).
Table 3 allows comparisons between
the observational constraints
retained in C95 and in this paper. The main
differences are for the luminosity values owing to the smaller
bolometric corrections and parallax we have used.
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B |
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V |
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(B-V) |
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BCV |
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this paper | Chmielewski et al. (1995) | |
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0.42 |
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Models have been computed using the CESAM code (Morel 1997).
About 600 mass shells describe each model; this number
increases up to 2100 for the model used in seismological analysis.
Around 400 and 30 models are needed to describe the evolutions of
Her A and B respectively.
Basically the physics employed is the same as in
Morel et al. (2000). The ordinary assumptions of stellar modeling are
made, i.e. spherical symmetry, no rotation, no magnetic field and no mass loss.
The evolutions are initialized with homogeneous zero-age main-sequence models
( ZAMS).
In the absence of satisfactory treatment of microscopic diffusion for
stars with mass larger than
,
we do not
take into account the diffusion of chemical species. This important
assumption is discussed in Sect. 5.
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Calibration without overshoot for ![]() |
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We have computed the evolution of models with
initial helium mass fraction, initial ratio of heavy elements to hydrogen,
masses and mixing length parameters respectively in the ranges
,
,
,
,
and
.
For the models of
HerA, as soon as
the evolution is halted.
Table 4 lists the set of "best'' modeling parameters
we adopt for the
Her A & B models
with and without overshooting of the convective
core of the primary. Figure 1 shows the corresponding
evolutionary tracks in the HR diagram. For the
calibration with core overshooting of
HerA,
we have not recomputed specific sets of models
to undertake another
minimization. We fit the
observable constraints within error boxes by adjustments of
and
,
the other modeling parameters having the values obtained without
overshooting. The larger value obtained for the age (
+250Myr)
results from the larger
amount of nuclear fuel available in the overshoot convection core.
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Figure 1:
Evolutionary tracks in the H-R diagram for models of ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
For our HerA model without overshooting, we have computed a
set of adiabatic frequencies of the normal modes
for degrees
in the frequency range 300 to
Hz.
The model corresponds to an evolved star which has burnt all the
hydrogen core and presents a radiative core and a convective
envelope starting at
,
(
). At age
Myr
the convective core which is present during the main sequence evolution
has disappeared about 440Myr ago. A zone of varying chemical
gradient was formed between the outer edge of the initial convective
core and the center. This
-gradient gives a rapid
variation of the sound speed and a large value of the maximum of
Brunt-Väisälä
frequency N (e.g. Unno et al. 1989):
Looking at the set of computed frequencies, we see that for each degree, the oscillation spectrum is no longer composed of two separated sets of modes with acoustic (p-modes) and gravity (g-modes) behavior as in solar-like stars but it is a complicated superposition of these two sets. Some of these waves have a dual character as they behave like pressure waves in the envelope of the star and gravity waves in the core. This greatly complicates the classification of the non-radial modes.
The distinction between the g- and p-modes
can be made by considering their normalized integrated kinetic energy
(or inertia)
(e.g. Christensen-Dalsgaard
& Berthomieu 1991).
The energy of the p-modes does not depend on the degree.
Figure 3 plots
this quantity as a function of the frequency for
.
The g-modes have
a much larger energy than the p-modes.
It appears that for
the modes all have both
energy in the core and in the envelop, so they are mixed modes.
Figure 4, upper panel, shows the distribution of
kinetic energy density of two of these modes along the radius of the star.
For
we
have a succession of modes with energy either in the core or in the
envelope with a few mixed modes as seen in Fig. 4, lower panel.
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Figure 2:
Propagation diagram: profiles as functions of the nondimensionned
stellar radius, of Brunt-Väisälä
frequency N and Lamb frequencies ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
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Figure 3:
Logarithm of the normalized integrated kinetic energy
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Open with DEXTER |
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Figure 4:
Density of kinetic energy
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Open with DEXTER |
These values are to be compared to the observations of
Martic et al. (2001) who derive a value around Hz
from the construction of an echelle diagram.
Figure 5 plots the large differences
with respect to frequency.
Except for modes
,
the points corresponding to high frequency modes
are close to the same flat curve around
Hz,
with oscillations due to the rapid variation of the adiabatic exponent
in the
helium ionization zone. As in the Sun, the g-modes have small amplitudes at
the surface and thus they will be hardly observable.
Therefore it is probable that the echelle diagram
used by the observers to extract the frequencies, taking
account of their asymptotic distribution, will work for
but
not for the mixed modes
.
The fit of the small differences:
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Figure 5:
Frequency differences between acoustic modes of
same degree and consecutive radial order as function of the frequency
(in ![]() |
Open with DEXTER |
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Figure 6:
Variation of the small differences d02 (full dots)
and
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Open with DEXTER |
The calibrated model of HerA calculated with overshoot
of its convective core,
has a larger radius inducing a smaller value of
as seen in Table 4.
We are aware that our models suffer from the absence of
diffusion of chemicals species. We do not introduce it as we do not have
a satisfactory description of the physical process which acts against
the too large efficiency of the gravitational settling in the envelope
of a main sequence stellar model with mass larger than
i.e. without a significant outer convection zone (e.g. Schatzman 1969;
Turcotte et al. 1998).
The derived metallicity of
HerA could not be
representative of
the initial mixture that formed
HerA & B. Because of the large
difference in
mass between the components, the diffusion has been more
efficient in
HerA than in
HerB
and therefore the adopted metallicity for
HerB needs to be increased.
As the mass of
HerB is close to the solar one, we can safely assume
that the change in metallicity has occurred at the same rate in the Sun and in
HerB. From ZAMS to the age
Myr, the metallicity of the Sun is increased by about 0.05dex.
Using Magain's (1987)
formula, an
increase corresponds to
an increase of
+75K in effective temperature.
Figure 1 shows that the locus (open square)
of
HerB in the HR diagram with
larger by 75K
is closer than before to the evolutionary tracks. To go further one needs
a measurement of the metallicity of
HerB.
With
and
it was not possible to obtain simultaneous satisfactory adjustments,
within the error boxes, for both components.
Realistic solutions are found with
.
That may
indicates that the suspected duplicity of
HerA is perhaps real.
In such a case the hypothetical unseen
component
Hera will be less massive than previously announced,
and will be a brown dwarf or a giant planet.
Though Table 4 exhibits insignificant small
differences between mixing length parameters of
HerA & B,
we obtained values of the order of unity,
as expected with the Canuto & Mazitelli (1991, 1992)
convection theory.
The differences between the C95
and our calibrations mainly result from the difference in distance.
In the HR diagram the locus of our
HerA model is found
in the Hertzsprung gap soon
after the main sequence, as expected for a sub-giant star.
In this part of the HR diagram, the effective temperature of a star model
varies rapidly with respect to time.
So the age is very sensitive to small changes of physics or of
modeling parameter. The small uncertainty we give for the
age corresponds to the crossing time of the error box in effective temperature.
It is valid only
with the physics we used and with the central values of other
modeling parameters.
In their study C95 obtained a value for the age:
Gyr close to ours. We emphasized the fact that
the crossing time of the C95 error box
in effective temperature (
K) is about twenty times smaller
than their estimated accuracy in age.
M | ![]() |
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age | R |
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Ag+ | 1.45 | 0.243 | 0.0269 | 3400 | 2.58 | 5763 | 41.6 |
Ag- | 1.45 | 0.243 | 0.0269 | 3372 | 2.53 | 5868 | 42.8 |
Am+ | 1.46 | 0.243 | 0.0269 | 3315 | 2.59 | 5811 | 41.6 |
Am- | 1.44 | 0.243 | 0.0269 | 3415 | 2.53 | 5808 | 42.7 |
Ay+ | 1.45 | 0.245 | 0.0269 | 3298 | 2.57 | 5808 | 41.8 |
Ay- | 1.45 | 0.242 | 0.0269 | 3332 | 2.55 | 5811 | 42.3 |
Az+ | 1.45 | 0.243 | 0.0274 | 3376 | 2.55 | 5810 | 42.4 |
Az- | 1.45 | 0.243 | 0.0264 | 3365 | 2.57 | 5808 | 41.8 |
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0 | 654.0 | 40.7 | 0.130 |
1 | 659.5 | 38.8 | 0.336 |
2 | 650.4 | 40.8 | 0.094 |
3 | 645.0 | 40.7 | 0.111 |
Table 5 lists the ages, the radii, the effective temperatures
and the large differences of HerA models
computed with modeling parameters within their error bars, but for mixing-length
parameters. The age of the model Ag- (respt. Ag+)
corresponds to the time elapsed from ZAMS
to the instant where the effective temperature crosses the left
(respt. right) limit of the error box in the HR diagram, namely
(respt.
).
The ages of other models listed in Table 5 corresponds
to the time elapsed from ZAMS
to the instant where the effective temperature crosses the central value
.
The variations of
reflect the differences in radius and mass of star models.
Table 5 shows that one derives
unrealistic precision for the modeling parameters
by defining their accuracy in
such a way that any combination of them, within their error bars,
will provide models of
HerA & B within
the observational constraints.
Owing to the weakness of the observing material we do not attempt to derive the
accuracy of the modelling parameters using the method developed by Brown et al. (1994).
A more sensitive modeling parameter
is the age due to the fast post main-sequence evolution of
HerA.
The accuracies of modeling parameters listed in
Table 4 may be optimistic. They mean that,
for a value of any modeling
parameter within its
interval of accuracy, one can find a set of other modeling
parameters, each of them within its own accuracy limit, in such a way that they
will provide models of
HerA & B within
the observational constraints.
The adiabatic oscillation spectrum of HerA is found to be a
complicated superposition of
acoustic and gravity modes. Some of these waves have a dual character.
This greatly complicates the classification of the non-radial modes.
For
the modes are mixed modes with both
energy in the core and in the envelope, they are mixed modes.
For
there is a succession of modes with energy either in the core or in the
envelope with a few mixed modes. This will have an implication for
the properties of the echelle diagram used by observers which will have a smooth
behavior only for the modes
.
The large difference
is found to be of the order of
Hz close to the
preliminary value derived from observations by Martic et al. (2001).
Six years ago, in the conclusion of their study,
Chmielewski et al. (1995) requested
"[] to go further, (i) a better photometry of component B and (ii)
the exact value of the parallax''.
Our work shows that the calibration of
Her remains a difficult task,
even with the disposal of improved observational data.
Among the binaries to be calibrated with some confidence,
Herculis is one of the most interesting owing to the difference of
evolutionary state of components resulting from their mass difference.
Though difficult,
Her is a reachable target for
modern spectroscopic and photometric apparatus.
Her
deserves interest to improve the accuracy of the modeling constraints hence,
of stellar models and seismological analysis. It is all the more desirable because
seismic observations of
HerA will hopefully give new constraints.
Acknowledgements
We thank M. Martic for having directed our attention to the opportunity to revisit the calibration of theHer binary system and to undertake the seismological analysis of the brightest component. We would like to express our thanks to the unknown referee, for helpful advises. This research has made use of the Simbad data base, operated at CDS, Strasbourg, France and of the WDS data base operating at USNO, Washington, DC, USA. This work has been performed using the computing facilities provided by the OCA program "Simulations Interactives et Visualisation en Astronomie et Mécanique (SIVAM)''.