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location of the actual detections of the object by Gaia. This computation
is based on a general simulation of asteroid detections by Gaia performed
by F. Mignard. In this plot, we assume the same set of parameters (axial
ratio, ecliptic longitude of the pole, sidereal period and rotational phase at
first detection) as used in producing Fig. 4 (left). As one can see, Gaia will
observe an object having the same orbit of 39 Laetitia many times (around
200 in this example, corresponding to the computed number of detections
in the field of view of the Gaia medium band photometer), and over a wide
interval of ecliptic longitudes. In order to compare this with a typical set
of ground-based data that have been sucient to derive a reliable estimate
of the asteroid pole, we show in the right panel of Fig. 5 an analogous plot,
showing the typical set of data (full lightcurves) used in the past to derive
the pole of asteroid 22 Kalliope.
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It is easy to see that Gaia sampling of the
longitude-magnitude plane in only 5 years of mission will be much denser
with respect to ground-based data obtained, in this particular example, by
means of observations spanning over an interval of 25 years. For reasons like
this, we expect that Gaia photometry will be a great resource for asteroid
science in the future.
Since the number of unknowns (the value of the spin period, two coordi-
nates for the pole axis orientation, a few parameters describing the shape,
and an initial rotation phase at
t
= 0 which has also to be determined) is
much smaller than the number of observations, it is in principle possible
to develop techniques of inversion of photometric data, to determine from
them the spin period, the spin axis orientation, and the overall shape. Dif-
ferent approaches are possible in principle, and have been independently
considered by different teams. In this paper, we explain in greater details
the algorithm developed by the Torino team.
In our approach, as a first approximation, we describe the shapes of the
objects by means of triaxial ellipsoids. This is certainly a major simplifica-
tion, but it has been found in the past to be reasonable for the purposes
of asteroid pole computation in a wide variety of cases.
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The shape param-
eters to be derived by photometry inversion are then the two axial ratios
b/a
and
c/a
. In addition to the rotation period, the pole coordinates, and
the unknown rotational phase at the zero-epoch, other parameters to be
derived are the linear slope of the magnitude-phase angle variation, which
is known to characterize the photometric behavior of real objects, and, at
a higher degree of complexity, the coecient of the linear increase of the
lightcurve amplitude as a function of phase angle. This effect, observed
for real asteroids,
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means that one must take into account that the linear
