We use in this section, the previous developments to study the error behavior of solar diameter measurements performed with a solar astrolabe in various seeing conditions and for different integration times of the CCD camera. It is useful to recall first the basic properties of the solar astrolabe experiment.
The diameter measurement principle consists in the determination of the time transits of the solar edges by the small circle localized at an altitude Z defined by the angle of the astrolabe prism (Laclare 1983). The solar diameter is deduced from the time interval between the two solar limb transits by the same small circle. Near the transit of the Sun by the small circle, the astrolabe gives two images of the object: a direct and a reflected ones. Each of them moves in the astrolabe focal plane due to the diurnal motion. The instant when the solar edge crosses the small circle corresponds to the contact of the two images. In the solar experiment, each solar image is formed on a CCD camera using a rotating shutter (Laclare & Merlin 1991). The solar image is analyzed by the acquisition system which extracts in real time, the solar edge. The edge is determined by the inflexion points of the solar limb function given by each image line.
For one solar image, the recorded data represent the solar edge associated to the corresponding time acquisition. For each solar edge transit by the small circle, data consist in a sequence of about 100 images recorded during about 20 seconds. The solar trajectory is determined from the data for both direct and reflected images. The intersection point of the trajectories gives the time when the solar edge crosses the small circle.
To simulate observations at the astrolabe, solar image sequences recorded in
various observation conditions are needed. For given observation conditions, a
simulated sequence is constituted with 50 direct images and 50 reflected ones.
Each simulated solar image has a size of 128 by 128''2.
The pupil aperture D and the observation wavelength are also
respectively equal to 10 cm
and . A Sun mean velocity of 15''
by second of time is taken in order to introduce the diurnal motion. To each
solar image is associated a recorded time taking in reference the first
simulated one. The acquisition sampling is fixed to 20 seconds. Each solar
image of the sequence is then analyzed and the solar edge extracted. As in the
solar experiment, the edge is defined as the zero of the second derivative of
each solar limb function assimilated to an image line. The edge associated to
the corresponding time allows to build the trajectories for both direct and
reflected solar images. The trajectories are after that approximated by
straight lines using least-square fits and the desired intersection point so
determined (Fig. 8).
Since the diameter measurement is deduced from the two instants of the solar
edge transit by the small circle, we will develop our error study around the
time transit behavior. Using the previous developments, many sequences of
solar images have been simulated in various observation conditions (Fried's
parameter r0, outer scale , exposure time t) (see
Sect. 2.3). The sequence analysis leads to the transit time of the Sun by
the small circle directly linked to the observation conditions. Thus, because
of temporal fluctuations observed on the trajectories, the time transit is not
well defined (Irbah et al. 1994). We define the error
associated to the transit time t0 as:
![]() |
(7) |
We will now analyze the error behavior with the observation conditions.
Solar image sequences composed of short and long exposure frames are used to
study the error behavior with the seeing. They are simulated for various
seeing conditions and supposing an infinite outer scale of the randomly
perturbed wavefronts. For a given Fried's parameter r0, the error is determined over an average of 50 transit times deduced from each kind of
the simulated solar sequences. The plot of the Fig. 9a shows the error
behavior with the seeing as obtained from the simulation. In each case, a
decrease of the error with good seeing conditions is observed. The use of long
exposure frames in the experiment gives however, a greater error on the time
contact than when using short exposure frames. This is better observed on the
figure for poor seeing conditions. Indeed, in case of short exposure frames,
the error is greater than 28 ms
when the r0 value is about
2 cm
and falls down to about 13 ms
for r0 equal to 5 cm
(see Table 1). The precision improvement is then equal to 54%. The error
still decreases to about
9 ms
for r0 equal to 8 cm
and improves the precision of about 68%. When using long exposure frames,
the error which is equal to
61 ms when r0 is equal to
2 cm,
falls down to about 20 ms
for r0 equal to 5 cm.
The precision improvement is then equal to 67%. The error decreases to
about 11 ms
for r0 equal to 8 cm
which gives a precision improvement of about 82%.
For mean seeing conditions at Calern Observatory (r0 = 4 cm) (Irbah et al. 1994), the error is about 16 ms if short exposure frames are used and about 29 ms in case of long exposure frames. According to the considered Sun mean velocity (see Sect. 3.2), the error on the solar diameter measurement is respectively equal to 0.5'' and 0.9'' when short and long exposure frames are used (see Table 2).
To study this effect, image sequences composed of short exposure frames
recorded in same seeing conditions are generated for different spatial
coherence outer scales. The error is determined over an average of
30 transit time samples. The Fig. 9b represents the time contact error
behavior with the outer scale
in two different seeing
conditions (r0=2 and 4 cm).
In each case, the error increases with the outer scale
in
order to reach a limit value. It is also observed from the figure that this
parameter has a significant effect only when it has small values
(
m).
It is limited after that by the error induced by the seeing. For the mean
seeing conditions of the Calern Observatory, the error on the time transit is
respectively equal to about 12 and 13 ms
for
equal to 2 and
8 m
(see Table 1). It remains then constant to about 14 ms
when
values are greater than 20 m.
This corresponds to an error on the solar diameter measurement of about
0.1''
when
takes values from 2 to 20 m (see Table 2).
We can notice finally that, although long exposure frames are obtained in best
seeing conditions, the precision is worse than that we obtain in case of poor
seeing conditions but with short exposure frames (with
1
and r0 = 2 cm,
ms while with
= 10 and
r0 = 4 cm,
ms). Thus, to have at least same accuracy's time contact than for poor seeing
conditions,
needs to be less than a factor 5. Thus, the
parameter appears to be fundamental in the solar astrolabe experiment. A
knowledge of time turbulence properties is then suitable to adjust
experimental parameters and improve accuracy of diameter measurement performed
with the solar astrolabe. Table 1 summarizes the results on the time contact
error obtained from this simulation while Table 2 presents the same results
but for the solar diameter
measurements.
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