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3 Error on the diameter measurements performed with a solar astrolabe. Results and discussion

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.

3.1 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.

  
\begin{figure}
\includegraphics [
height=6.4317cm,
width=15.1853cm
]{ds1705f9.eps}\end{figure} Figure 9: Time contact error with: a) the seeing (-*: short and .$\bullet.$: long exposure frames), b) the outer scale $\mathit{L}_{0}$ (-*: r0= 2 cm and .$\bullet.$: r0 = 4 cm) and c) the parameter $R_{\tau}$ (-*: r0 = 2 cm and .$\bullet.$: r0= 4 cm) (see text)

3.2 Simulation of solar data recorded in various observation conditions

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 $0.55\,\mu{\rm m}$. 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).

3.3 Results and discussion

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 $\mathit{L}_{0}$, 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 $\Delta t$associated to the transit time t0 as:
\begin{eqnarray}
\Delta t=\left\vert t_{0}-t_{\rm true}\right\vert \end{eqnarray} (7)
where $t_{\rm true}$ denotes the true contact time of solar images obtained in ideal seeing conditions.

We will now analyze the error behavior with the observation conditions.

3.3.1 Error due to the seeing

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 $\Delta t$ 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).


 
Table 1: Error on the time contact

\begin{tabular}[c]
{\vert c\vert c\vert c\vert c\vert c\vert c\vert c\vert c\ver...
 ... 107.8 & 97.4\\ & & & 40 & 24.6 & 13.7 & 80 & 108.0 & 98.5\\ \hline\end{tabular}


 
Table 2: Error on the solar diameter measurements

\begin{tabular}[c]
{\vert c\vert c\vert c\vert c\vert c\vert c\vert c\vert c\ver...
 ...4 & 70 & 3.2 & 2.9\\ & & & 40 & 0.7 & 0.4 & 80 & 3.2 & 3.0\\ \hline\end{tabular}

3.3.2 Error due to the spatial coherence outer scale of the perturbed wavefronts

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 $\Delta t$ is determined over an average of 30 transit time samples. The Fig. 9b represents the time contact error behavior with the outer scale $\mathit{L}_{0}$ in two different seeing conditions (r0=2 and 4 cm). In each case, the error increases with the outer scale $\mathit{L}_{0}$ 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 ($\mathit{L}_{0}\leq8$ 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 $\mathit{L}_{0}$ equal to 2 and 8 m (see Table 1). It remains then constant to about 14 ms when $\mathit{L}_{0}$ values are greater than 20 m. This corresponds to an error on the solar diameter measurement of about 0.1'' when $\mathit{L}_{0}$ takes values from 2 to 20 m (see Table 2).

3.3.3 Error due to the exposure time relatively to the observation conditions

A similar study has been also developed using sequences simulated in case of different exposure times relatively to the turbulence characteristic time. The outer scale is supposed infinite and two seeing conditions are considered (r0=2 and 4 cm). We define the parameter $R_{\tau}$ as the ratio between the exposure time and the characteristic time. The Fig. 9c shows the time difference $\Delta t$ behavior with the $R_{\tau}$ parameter. For the considered seeing conditions, it is observed a growth of the time difference $\Delta t$ for small $R_{\tau}$ values which reaches a limit value. For the mean seeing condition of Calern Observatory, the error on the time transit is equal to 15.5 ms (0.5'' on the diameter measurements) for $R_{\tau}$ equal to 1 (short exposure frames) which corresponds to the error due to the seeing (r0 = 4 cm). It is multiplied by a factor 2.8 then 4.2 when $R_{\tau}$ value goes from 10 to 20. It remains to a constant value of about 100 ms (3'' on the diameter measurements) for the higher values of the $R_{\tau}$parameter. Consequently, the precision loss is about 280% when $R_{\tau}$goes from 1 to 10 and about 420% when it becomes equal to 20 (see Tables 1 and 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 $R_{\tau}$ $\leq$ 1 and r0 = 2 cm, $\Delta t= 28 $ ms while with $R_{\tau}$ = 10 and r0 = 4 cm, $\Delta t = 43 $ ms). Thus, to have at least same accuracy's time contact than for poor seeing conditions, $R_{\tau}$ needs to be less than a factor 5. Thus, the $R_{\tau}$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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