New determination of the size and bulk density of the binary asteroid 22 Kalliope from observations of mutual eclipses

Abstract

In 2007, the M-type binary asteroid 22 Kalliope reached one of its annual equinoxes. As a consequence, the orbit plane of its small moon, Linus, was aligned closely to the Sun's line of sight, giving rise to a mutual eclipse season. A dedicated international campaign of photometric observations, based on amateur-professional collaboration, was organized and coordinated by the IMCCE in order to catch several of these events. The set of the compiled observations is released in this work. We developed a relevant model of these events, including a topographic shape model of Kalliope refined in the present work, the orbit solution of Linus as well as the photometric effect of the shadow of one component falling on the other. By fitting this model to the only two full recorded events, we derived a new estimation of the equivalent diameter of Kalliope of 166.2+/-2.8km, 8% smaller than its IRAS diameter. As to the diameter of Linus, considered as purely spherical, it is estimated to 28+/-2 km. This substantial "shortening" of Kalliope gives a bulk density of 3.35+/-0.33g/cm3, significantly higher than past determinations but more consistent with its taxonomic type. Some constraints can be inferred on the composition. Comment: 59 pages, 8 tables, 15 figures submitted to Icarus

Full text

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New determination of the size and bulk density of the binary asteroid 22
Kalliope from observations of mutual eclipses
P. Descamps1, F. Marchis1,2,11, J. Pollock3, J. Berthier1, F.Vachier1, M. Birlan1, M. Kaasalainen4,
A.W. Harris5, M. H. Wong2, W.J. Romanishin6, E.M. Cooper6, K.A. Kettner6, P.Wiggins7, A.
Kryszczynska8, M. Polinska8, J.-F. Coliac9, A. Devyatkin10, I. Verestchagina10, D. Gorshanov10
1 Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, UMR8028
CNRS, 77 av. Denfert-Rochereau 75014 Paris, France
2 University of California at Berkeley, Department of Astronomy, 601 Campbell Hall, Berkeley, CA
94720, USA
3 Appalachian State University, Department of Physics and Astronomy, 231 CAP Building, Boone, NC
28608, USA
4 Department of Mathematics and Statistics, Gustaf Hallstromin katu 2b, P.O.Box 68, FIN-00014
University of Helsinki, Finland
5 DLR Institute of Planetary Research, Rutherfordstrasse 2, 12489 Berlin, Germany
6 University of Oklahoma, 440 West Brooks, USA
7 Tooele Utah 84074-9665, USA
8 Astronomical Observatory, Adam Mickiewicz University, Sloneczna 36, 60-286 Poznan, Poland
9 Observatoire de la Farigourette 13012 Marseille
10 Central Astronomical Observatory, Pulkovskoe chaussee 65/1, 196140 St.-Petersburg, Russia
11 SETI Institute, 515 N. Whisman Road, Mountain View CA 94043, USA
Pages: 59
Tables: 8
Figures: 15
Corresponding author:
Pascal Descamps
IMCCE, Paris Observatory
77, avenue Denfert-Rochereau
75014 Paris
France
descamps@imcce.fr
Phone: 33 (0)140512268
Fax: 33 (0)146332834
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Abstract
In 2007, the M-type binary asteroid 22 Kalliope reached one of its annual equinoxes. As a
consequence, the orbit plane of its small moon, Linus, was aligned closely to the Sun's line of
sight, giving rise to a mutual eclipse season. A dedicated international campaign of photometric
observations, based on amateur-professional collaboration, was organized and coordinated by
the IMCCE in order to catch several of these events. The set of the compiled observations is
released in this work. We developed a relevant model of these events, including a topographic
shape model of Kalliope refined in the present work, the orbit solution of Linus as well as the
photometric effect of the shadow of one component falling on the other. By fitting this model to
the only two full recorded events, we derived a new estimation of the equivalent diameter of
Kalliope of 166.2±2.8km, 8% smaller than its IRAS diameter. As to the diameter of Linus,
considered as purely spherical, it is estimated to 28±2 km. This substantial “shortening” of
Kalliope gives a bulk density of 3.35±0.33g/cm3, significantly higher than past determinations
but more consistent with its taxonomic type. Some constraints can be inferred on the
composition.
Keywords
Asteroids, rotation, surfaces – satellites of asteroids - eclipses - photometry
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1. Introduction
Ten years ago, Pravec and Hahn (1997) suspected that the near-Earth asteroid 1994 AW1 could
be a binary system from the analysis of its complex ligthcurve revealed in January 1994. Since
that claim, a large number of binary systems have been discovered in all populations of minor
bodies in the solar system (Noll, 2006). Extensive and systematic adaptive optics (AO)
astrometric follow-up of all known large binary asteroids was performed by our group in order
to improve moonlet orbits (Marchis et al., 2003a, 2004). We published a preliminary solution
for the orbit of Linus (Marchis et al., 2003b), satellite of the large (DIRAS = 181 km, Tedesco et
al., 2002) main belt asteroid 22 Kalliope (Margot et al, 2001). Further AO astrometric
observations were carried out up to December 2006, just before the season of mutual events,
leading to a significantly corrected orbit solution (Marchis et al., 2008a). Thanks to this updated
knowledge of the orbit of Linus, eclipse events were predicted to occur for this binary system
from late February through early April 2007 (Descamps et al., 2006). The observation of such
events requires a favourable geometry, when the Earth is near the satellite’s orbital plane so that
eclipses occur at regular intervals. Photometric observation of mutual events is a powerful
method to detect and study small asynchronous binaries (Pravec et al. 1998, Mottola and
Lahulla, 2000, Ryan et al., 2004, Pravec et al., 2006) as well as doubly synchronous pairs of
twin asteroids (Kryszczyńska, 2005, Behrend et al., 2006, Descamps et al., 2007, Marchis et al.,
2007a, Kryszczyńska et al., 2008).
Among all asynchronous main belt binary asteroids, the secondary-to-primary size ratio of the
22 Kalliope system is the highest with a value estimated to 0.2 which is considered as the lower
photometric detection limit of a binary system in mutual eclipse (Pravec et al., 2006). At the
present time, the size of Linus is roughly bound between 20 and 40km, based on measurement
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of the secondary to primary flux ratio from adaptive optics imaging (Margot and Brown, 2003,
Marchis et al., 2003b). With an apparent size of about 25 milliarcseconds (mas), Linus always
remains beyond the resolving power of the largest Earth-based telescopes (which is for instance
of 55mas for the Keck-10m telescope). With the 2007 favourable circumstances, photometric
observations of mutual eclipses enabled us to tightly constrain the physical characteristics of
both Linus and Kalliope.
Before tackling the issue of detection of mutual eclipses within the photometric observations
collected in 2007 in section 3, we strive in the first place to get an improved polyhedral shape
model of Kalliope in section 2, required for a fair modelling of the events. The comprehensive
model is fully described in section 4 and fitted to the observations (section 5). The validity of
our solution is tested in section 6 with the report of an observation of a stellar occultation of
Kalliope and Linus on November 2006. Lastly, we address in sections 7 and 8 some issues as
regards the structure and composition of Kalliope and the origin of this binary main-belt asteroid
system.
2. Improving the topographic shape model of Kalliope
The shape and pole solution are crucial parameters to predict and account for observed
photometric rotational lightcurves. Kaasalainen et al. (2002a) derived an initial convex
polyhedral shape solution of 22 Kalliope from their noteworthy lightcurve inversion method
(Kaasalainen et al., 2001). This model should be considered as a realistic approximation of its
highly irregular shape. From November 2006 to March 2007, a long-term photometric follow-up
of Kalliope was performed with the 0.4m telescope at Appalachian State University’s Rankin
Science Observatory, located in western North Carolina. Images were taken in the R band using
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an SBIG ST-9e CCD camera and the data were reduced by aperture photometry. The collected
lightcurves are displayed on Figure 1. This new set of lightcurves supplied the photometric
database of Kalliope in order to update the shape, pole and period solution using the
aforementioned inversion method. We took advantage of the edge-on aspect of Kalliope, never
observed yet, to improve the current shape model. The derived spin vector solution in J2000
ecliptic coordinates is λ=195±3° and β=+7±2° with a sidereal spin period of
4.148199±0.000001hours. The resulting asteroid model is represented as polyhedrons with
triangular surface facets. The pear-shaped model for Kalliope is rendered in Figure 2. It turns out
to be quite elongated and flattened in a conic-like manner with a theoretical dynamical flattening
J2 of 0.19, assuming a uniformly dense body. The scattering of solar light from the surface is
synthesized considering the empirical Minnaert’s law (1941) with a limb darkening parameter k
adjusted to 0.53±0.02, typical of atmosphereless bodies. The surface photometric function is an
important determinant of the amount of contrast (darkening) for a given topography. Although
the limb-darkening parameter slightly depends on the phase angle, the photometric function of
Minnaert provides a reasonable working approximation to more complex, physically motivated
models such as the Hapke’s function (1981). Nevertheless it becomes somewhat irrelevant at
phase angles larger than ~20° (Veverka et al., 1978). Since the present inversion method
analysis is mainly based on near-opposition lightcurves and does not take into consideration the
modification of the scattering properties with the phase angle, small discrepancies between
observed and modelled lightcurves may arise at this range of significant phase angles. In Figure
1, synthetic lightcurves were generated and superimposed on the observations. The evolving
peak-to-peak amplitude together with the overall shapes of lightcurves are well accounted for to
within about 0.01mag, let alone the lightcurves taken from early March 2007. At this epoch the
phase angle was greater than 20° and the departure between our model and the observations
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reaches 0.02mag at most (effect of such a discrepancy on event modelling is discussed in section
4.1). Furthermore, a posteriori simulations made including the satellite Linus do not show any
significant effect, larger than 0.007mag, on the lightcurves (see in Fig. 1 the dash-dotted curve
of the March 25 observation). Thereby, neglecting Linus during the lightcurve inversion process
does not affect the resulting shape but rather slightly underestimates the limb-darkening
parameter by artificially reducing the true amplitude of Kalliope.
3. The 2007 mutual events
3.1 General description
As the orbit of Linus is nearly located in the equatorial plane of 22 Kalliope, the system
undergoes seasons of mutual eclipses and occultations at its equinoxes. This configuration,
which occurs every 2.5 years, happened in the northern hemisphere spring of 2007. The
observation of such events provides opportunities for very precise astrometry of the satellite and
studies of its physical characteristics that are not otherwise possible.
Due to the fast-evolving aspect of the system, as seen by an Earth observer, the season of mutual
events lasts only three months. As a result of the distance of 22 Kalliope from the Earth (> 2
AU) and its axial tilt to the ecliptic plane of nearly 90°, mutual eclipses took place in February
2007 and lasted until early April 2007. The proximity of Kalliope to the Sun in May made
observations of the mutual occultations difficult.
The brightness of Kalliope (mv = 11) permitted photometric observations with a small aperture
telescope. Anomalous attenuation events were predicted to last about 1 – 3 hrs with detectable
amplitude ranging from 0.03 to 0.08 magnitudes (Descamps et al., 2006). The magnitude drop
during a total eclipse of Linus, the smaller of the two, mainly depends on the relative sizes. An
eclipse of Kalliope by Linus is always partial and the decrease in luminosity will also depend on
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the shape of Kalliope. We estimated that a photometric accuracy of about 0.01-0.02mag was
necessary to detect such small photometrical variations.
3.2 Observations
The number of observed events depends greatly on the number and geographical distribution of
available observers. This is the reason for which an international campaign of observation of
these events was set up by the IMCCE team coordinating the efforts to gather observations of as
many as possible events1. The network of observers is given in the Table 1. Table 2 lists the
collected observations which are plotted on Figure 3. Three positive detections have been
recorded of which only one was a total eclipse of Linus on March 8th. For the eclipse of Kalliope
on March 17th, the ingress was observed at Haute-Provence observatory, France while the egress
was recorded in Oklahoma, USA. We merged these two observations into one unique event.
Finally, only two complete events were observed. These data are important to assess the
durations and thereby constrain the sizes. A third event, an eclipse ingress of Kalliope by Linus,
was observed on March 27th, but was not used in the analysis because its incompleteness.
An eclipse can be identified unambiguously if the two components of the binary system have a
similar size, but in the case of a small satellite, such as Linus of Kalliope, the effect is so subtle
that it may be confused with the primary’s lightcurve. To overcome this difficulty, two
lightcurves taken a few nights apart are superimposed and subtracted from each other. We will
get a positive detection if the residual curve, called the drop curve, reveals a clear light
attenuation imputable to a supposed event. The point-to-point subtraction is performed after
expanding one of the two lightcurves in Fourier series and interpolating it at the times of the
other. Figures 4-6 shows the positive detections. Lightcurves recorded during two closely spaced
1
http://www.imcce.fr/page.php?nav=en/observateur/campagnes_obs/kalliope/index.php
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sessions are overlapped and plotted on the left side of each figure. The corresponding magnitude
drop curve, resulting from the subtraction from each other, appears on the right side. Despite the
unavoidable presence of noise on the data, prominent light variations are undoubtedly detected
and reveal ongoing mutual events within the system of Kalliope.
Table 3 summarizes the characteristics of the detected events. For each of them, magnitude drop
and times of observed eclipse ingress and egress are reported which may be compared with the
expected times, computed from our last orbit solution given in Table 4 (Marchis et al., 2008).
The discrepancies are large and may amount to ~2h (180km in position). This is due to the orbit
model itself whose the global 1σ RMS error is on the order of 40 mas (70km).The reason lies in
a simple preliminary keplerian model used to account for the AO astrometric positions.
3.3 Derivation of the size ratio between Linus and Kalliope from the total eclipse of Linus
We can take advantage of the full observation of the total eclipse of Linus by Kalliope to
estimate the relative sizes of either component of the system. During a total eclipse of Linus,
scattered solar light of Linus is not observed on Earth so that the light attenuation is only
correlated to the relative size of Linus with respect to Kalliope. The depth, expressed in
magnitudes, of the attenuation is related to the ratio of cross-diameters of either component by
the simple formula:
()
2
1 log 5 . 2r
F
F
t
s
+=
(1)
where r is the secondary-to-primary size ratio, Fs the flux of the secondary, and Ft the total flux.
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If we apply this formula for an observed attenuation of 0.05mag at the time of disappearance of
Linus (Fig.4), assuming the IRAS diameter of Kalliope of 180±4.5km (Tedesco et al., 2002), we
get a size ratio of 0.21 corresponding to a size of Linus of 38km. This is in agreement with the
size ratio of 0.2, inferred from the flux ratio measurement in the AO observations (Margot and
Brown, 2003, Marchis et al., 2003b). Nevertheless, the reliability of such a size determination
depends to a great extent on the adopted size of Kalliope. The mutual events observations
provide a unique opportunity to get an independent measurement of the very size of Kalliope
which will mainly decide on the duration of an event. Furthermore, if we look at the drop
curves, we must pay attention to the fact that they do present neither the same pattern nor similar
amplitudes. They are not at all reminiscent of the reversed bell curves observed in the case of
classical mutual events within planetary satellite systems. Quite obviously, the shape of Kalliope
plays a key role in the way the light attenuation is taking place. Therefore, we should first
address the modelling of an eclipse phenomenon, involving a non-spherical primary body, in
order to derive a trustworthy new assessment of the size of Kalliope.
4. Modelling the mutual eclipses
4.1 Description of the synthetic model
From the study presented in section 2, we have a precise knowledge of the rotation and shape of
the primary. The shape of the secondary cannot be constrained by mutual eclipse observations
owing to its smallness relatively to the primary and will be consequently considered as purely
spherical. Another point of importance lies on the precise localization of the shadowed area on
the surface of Kalliope whenever it is eclipsed by Linus. This is achieved if we conveniently
consider that an eclipse is an occultation from the standpoint of a Sun observer. From the Sun,
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the cross sections of the “occulting” and “occulted” bodies draw outlines. Facets of the
“occulted” body (the eclipsed body for an Earth observer) which are located inside or on the
border of the occulting body outline do not receive any solar light and their scattered flux is
equal to zero. After going back to the initial frame facing the observer – which is that of the
tangent plane of the observation – we have to retrieve the eclipsed facets and to carry out a
summation of the scattered flux over all visible facets including the eclipsed facets. In our model
the penumbral annulus is not taken into account because each body is located very close from
each other so that the penumbral width is negligible. Light travel time between each body is
likewise neglected. The model provides the values of the quantities F1e, F2e, F1 and F2 which stand
for the fluxes of Kalliope (subscript 1) and Linus (subscript 2) in and out eclipse. The magnitude
drop of a partial eclipse of Kalliope by Linus (2E1) or of an eclipse of Linus by Kalliope (1E2)
is straightforwardly provided by the following formula:
F1E2=F1F2
e
F1 F2
(2)
F2E1=F1
eF2
F1 F2
(3)
The eclipse of Linus is said total if F2e=0. Figure 7 shows the apparent configuration of the
system generated at three times of the partial eclipse of Kalliope by Linus on March 17th. The
dark area on Kalliope, which is the shadow of Linus, covers an irregular region, slightly tilted
over the cylindrical shadow. Thus the umbra obliquely falls upon the surface, causing, for an
Earth-based observer, a distortion of the shadow which tends to encompass a much larger area
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than we would have on a purely spherical Kalliope. On the other hand, the irregular shape of
Kalliope makes that its cross-section strongly departs from that corresponding to its effective
diameter. Accordingly, the part of flux which is removed from the total collected light is
enhanced which explains why we observe decrease in magnitude as large as 0.08mag when the
shadow of Linus is falling on Kalliope.
We saw in section 2 that our updated shape model of Kalliope was able to reproduce observed
lightcurve to within about 0.02mag. A question which may then come to mind is whether this
topographic model is capable of simulating attenuations as faint as 0.05-0.08 magnitudes. Let
δF1, much less than F1, be the error in the Kalliope flux estimation due to its approximated shape.
Take now F1+δF1 and F1e+δF1, and substitute for F1 and F1e into Eq. [3] (same reasoning with the
Eq. [2]). We have

F2E1=F1
e F1 F2
F1 F1 F2
=F2E11 F1
F1 −F1
e
eF2O F1
F1 F2F1
2
F1
2  (4)
Hence, assume that F1e ≃F1-δF1 and F2 ≤F1

F2E1≈F2E11  F1
2
F1
2 O F1
2
F1
2  (5)
Taking into account that the error expressed in magnitude is given by
2.5log(1+δF1/F1)~0.02mag, the error in the magnitude drop then amounts to ~2.5log(1+δF12
/F12)~0.0004mag, which proves the ability of our shape model to account for the drop curve
during an event. As a rule, if we want to have a fair representation of the magnitude drop, it is
sufficient to have a shape approximation not yielding photometric departure from the observed
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lightcurves, no matter what the aspect and phase angles, by more than ~0.05mag (yielding
0.002mag on the drop curve). This condition rules out a trivial triaxial ellipsoid as a valid shape
model to simulate the light loss during an event involving an irregular body.
4.2 Photometric effect of the eclipse parameters
This part is aimed at scrutinizing the sensitivity of the drop curve against the main parameters
involved in an eclipse event. The free parameters taken into account are the size of bodies, the
orbital pole, the shape model of Kalliope and the Minnaert limb-darkening parameter. Despite
their unavoidable correlations, this study is undertaken to highlight the characteristic dominant
effect of each parameter. It is carried out with the event on March 8th. Apart from our refined
topographic shape model, we considered another shape model provided by its current ellipsoidal
figure (a/b=1.33, b/c=1.27, de Angelis, 1995). The effects on the duration, the amplitude and the
shape of the drop curve of the event are visible on Figure 8.
Fig. 8F shows that the most conspicuous effect arises from the shape model of Kalliope. For a
same orientation and effective diameter, the ellipsoidal model cannot account not only for the
irregular profile of the drop curve but also for the amplitude which is then significantly
underestimated. In other words, using an ellipsoidal model will yield an overestimation of the
Linus size, required to compensate the lack of amplitude. As to the influence of the Minnaert
limb-darkening parameter (Fig. 8E), it is negligible given our estimated value of 0.53±0.02 (see
section 2).
Based on these results, we will proceed in the rest of this study by definitely adopting the
polyhedral shape solution of Kalliope obtained in section 2. As far as the size effects are
concerned, they are consistent with what it is expected in terms of amplitude and duration
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variations, namely, the smaller the Kalliope size, the shorter the duration and the weaker the
amplitude (Fig. 8A). However, the Linus size has no measurable effect on the duration, but only
on its amplitude (Fig. 8C & 8D). Basically, the most significant changes in the shape of the drop
curve come from the orbital pole orientation. The influence of each ecliptic coordinate has been
explored separately and the prominent effect is decidedly on the very form of the drop curve
which may markedly evolve with only slight variations on either coordinate.
Lastly, from this series of trials, we can now draw some general rules to iteratively process each
observed event and achieve a rough starting solution for each parameter:
1. Determining the best Linus orbital plane orientation able to accurately match the general
form of the drop curve.
2. Fitting the size of Kalliope to the duration.
3. Fitting the size of Linus to the amplitude.
5. General solution
5.1 Physical solution for Kalliope and Linus
Practically, we perform a full grid analysis over the multidimensional space of the free
parameters (ecliptic coordinates of the orbit pole, equivalent radii of Kalliope and Linus) using a
goodness-of-fit criterion taken as the averaged differences between simulations and
observations. The criterion Θ is defined as:
()
n
CO
mag
n
i
ii
∑
=
−
=Θ
1
2
)(
(6)
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Where n is the data number, and Oi and Ci are observed and calculated magnitudes.
Before each fit, the spin axis of Kalliope was finely determined from the corresponding
reference lightcurve. We found the following J2000 ecliptic coordinates, (192±2°, +7±1°) on
March 6th and (196±2°, +6±1°) on March 18th. The grid search in the parameter space was
performed inside a region surrounding an initial guess of the best solution provided by the three
rules described in the previous section. The best solution is reached for the event on March 8th
by minimizing the goodness-of-fit criterion with Θ=0.008mag. Then we derive a precise
estimate of the uncertainty on the Kalliope effective diameter by investigating the relevance of
the event duration to the size of Kalliope (Fig. 9). Given an accuracy of 1 minute of time on the
ingress and egress times determination (Table 3), i.e. 2 minutes in the duration assessment, we
derive an effective diameter of Kalliope of 166.2±2.8km which turns out to be approximated by
a triaxial ellipsoid with semi-major axes a=117.5km, b=82km, c=62km. The Linus diameter is
then fitted to 28±2km. With this new value of the effective diameter of Kalliope, we derive a
visible albedo pv =0.172, higher than its previous value of 0.12, for an absolute magnitude
H=6.45. This albedo is in agreement with the mean albedo of M-type asteroids (Belskaya and
Lagerkvist, 1996). In order to ascertain the reliability of these results, we successfully fitted the
event on March 17th with the same size parameters and Θ=0.014mag, giving a high degree of
coherence and confidence to our size derivation. The best-fitted drop curves are displayed on
Figure 10. As for the Linus orbital plane orientation, their J2000 ecliptic coordinates are fitted to
(194±1°, -4±1°) on March 8th and (198±1°, -5±1°) on March 17th. Although it implies a moderate
inclination over the spin axis of Kalliope of ~11°, yielding an expected nodal precession rate of
0.15°/day. The physical interpretation of such a motion in the spin and orbital axes should be
2
If the size D of an asteroid is known and its absolute magnitude H, its geometric visible albedo pv can be
1329
)(
v p
derived from
5/
10.
H
kmD
−
=

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cautiously considered and deserves further observational confirmations and theoretical
investigations.
It is worth noting that we get a new secondary-to-primary size ratio of 0.169±0.006, instead of
0.21 derived from AO observations. This lower size ratio is likely due to the fact that Kalliope is
always resolved in AO images and consequently its flux is spread over its apparent size,
differently than it would be with a point-like stellar profile. A consequence of that photometric
bias is the overestimation of the secondary size if we simply adopt the previously published
IRAS diameter for the primary. Besides, in the case of an irregular body such as Kalliope for
instance, the expected theoretical magnitude drop, given by Eq. [1], is basically inadequate to
give a fair idea of the real level of the light attenuation. With the new size ratio, the decrease in
magnitude assuming spherical primary would be but 0.03mag, lower than what was really
observed by a factor of ~3. This difference underlines the enhancement of light loss due to the
degree of non-sphericity of the primary. As a consequence, binary systems with much smaller
size ratio than 0.2 may be studied through mutual events observations. Adopting a detection
threshold of ~0.02mag, we may infer a limiting secondary-to-primary size ratio of ~0.07,
necessary to detect mutual events within a binary asteroidal system.
5.2 Astrometry of the events
It is straightforward from our photometric model to infer ancillary astrometric relative positions
of Linus in the tangent plane of the observation at the times of eclipse ingress and egress. The
astrometric positions labelled in Table 5 are extremely accurate with a typical error of 1 mas and
4 mas in X and Y axes respectively. A significantly more advanced dynamical model, taking
into account the non-spherical gravitational field of the primary as well as the moderate
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