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Subsections

2 Synthetic images as observed at the solar astrolabe

In this section, solar images with similar characteristics as those recorded at the astrolabe through the atmospheric turbulence are simulated. The optical response of the whole system atmosphere and instrument is then calculated using a fractal model to generate the randomly perturbed wavefronts. It is used to build solar images in various seeing conditions and with different exposure times. Let us first present the synthetic image of the Sun which is needed for the simulation.

2.1 Synthetic image of the Sun

Several models of the solar limb darkening function exist. The model we use to build the synthetic image of the Sun is given by (Allen 1973):
\begin{eqnarray}
O_{\lambda}(\alpha)=O_{\lambda}(0)\left[ 1-u_{2}-v_{2}+u_{2}\cos
(\alpha)+v_{2}\cos^{2}(\alpha)\right] \end{eqnarray} (1)
where $O_{\lambda}(\alpha)$ is the intensity of the solar continuum at an angle $\alpha$ from the centre of the disk. $\alpha$ represents the angle between a Sun's radius vector and the line of sight. $\alpha=0$ corresponds to the centre of the disk and $\alpha=$ $\frac{\pi}{2}$ to the solar edge. The ratio $\frac{O_{\lambda}(\alpha)}{O_{\lambda}(0)}$, which varies with the wavelength $\lambda$ defines the limb darkening. For the used observation wavelength ($\lambda=0.55\ \mu$m), the constants u2 and v2 are respectively equal to 0.93 and -0.23. The plot in Fig. 2a shows the solar limb darkening function where the intensity was normalized between 0 and 255 for our needed calculations. The Fig. 2b represents a solar image sample where the field size is equal to 128 by 128''2.

  
\begin{figure}
\includegraphics [height=1.7331in,width=3.2396in]{ds1705f2.eps}\end{figure} Figure 2: Solar limb darkening function a) and the synthetic image of the sun b)

2.2 Simulation of the optical response of the whole system, atmosphere and instrument using a fractal model

The psf of the whole system instrument and atmosphere is needed to generate the solar image sequences as recorded through the atmosphere in various observation conditions. We present first, the fractal model used to generate the two-dimensional randomly perturbed wavefronts and then the deduced optical responses.

2.2.1 The randomly perturbed wavefront synthesis using a fractal model

The turbulence follows a fractal structure. The model used to generate the randomly perturbed wavefront function $\varphi$ is based on a mid point displacement algorithm (Lane et al. 1992; Beaumont et al. 1996). To simulate a L by L phase screen, the first step is to choose three points $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ disposed at the summits of an isoceles triangle with base and height equal to 2L (Fig. 3). Phase function values at these points are Gaussian random variables with zero means and with variances given by:
\begin{eqnarray}
\sigma^{2}\left( \phi_{1}\right) =D_{\phi}\left( \sqrt{5}L\right)
-\frac{1}{2}D_{\phi}\left( 2L\right) \end{eqnarray} (2)
and
\begin{eqnarray}
\sigma^{2}\left( \phi_{2}\right) =\sigma^{2}\left( \phi_{3}\right)
=\frac{1}{2}D_{\phi}\left( 2L\right) \end{eqnarray} (3)
where $D_{\phi}\left( r\right) =\left\langle \left\vert \varphi(x)-\varphi
(x+r)\right\vert ^{2}\right\rangle = 6.88\left(\frac{r}{r_{0}}\right)
^{\frac{5}{3}} \ast$
\begin{eqnarray}
\left[1 -1.485\left( \frac{r}{\mathit{L}_{0}}\right)
^{\frac{1}...
 ...6.281\left(
\frac{r}{\mathit{L}_{0}}\right) ^{\frac{7}{3}}\right] \end{eqnarray} (4)
$< \,\gt$denotes a mean average. $D_{\phi}\left( r\right) $ is the phase structure function deduced from the Kolmogorov's law and the Von Karman model (Ziad 1993). r0 is the Fried's parameter which qualifies the observations (seeing) (Fried 1966) and $\mathit{L}_{0}$ the spatial coherence outer scale of the wavefronts (Borgnino 1990).

  
\begin{figure}
\includegraphics [
height=2.38in,
width=2.8634in
]{ds1705f3.eps}\end{figure} Figure 3: The fractal interpolation grid used to generate the perturbed wavefronts

Equations (2), (3) and (4) show that $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$follow the theoretical phase structure function deduced from the Kolmogorov turbulence law since:
\begin{align}
&\left\langle \left\vert \phi_{1}-\phi_{2}\right\vert ^{2}\right\r...
 ...)\right\vert ^{2}\right\rangle =D_{\phi}\left(
2L\right).\tag*{(5.3)}\end{align}
Three new points $\phi_{12}$, $\phi_{13}$ and $\phi_{23}$ of the phase screen are generated midway between each existing points by a process of linear interpolation and addition of independent random variables $\varepsilon_{1}$,$\varepsilon_{2}$ and $\varepsilon_{3}$, such as:
\begin{align}
&\phi_{12}=\frac{1}{2}\left( \phi_{1}+\phi_{2}\right) 
+\varepsilo...
 ...ac{1}{2}\left( \phi_{2}+\phi_{3}\right) 
+\varepsilon_{3}\tag*{(6.3)}\end{align}
with variances given by:
\begin{displaymath}
\sigma^{2}\left( \varepsilon_{1}\right) 
=\sigma^{2}\left( \...
 ...}{2}L\right) -\beta D_{\phi
}\left(\sqrt{5}L\right)\tag*{(7.1)}\end{displaymath}
and
\begin{displaymath}
\sigma^{2}\left( \varepsilon_{3}\right) 
=D_{\phi}\left( L\right) -\beta
D_{\phi}\left( 2L\right)\tag*{(7.2)}\end{displaymath}
where $\beta$ is a semi-empirical coefficient equal to 0.25.

The iterative procedure interpolation/displacement is then repeated in each new generated triangle until the desired numbers of samples are obtained. The needed L by L phase screen is extracted from the middle of the base of the original isoceles triangle (Fig. 3). The Fig. 4 shows the simulation of a perturbed wavefront sample using the model. It is simulated for a Fried's parameter r0 equal to 4 cm and a spatial coherence outer scale supposed infinite.

  
\begin{figure}
\includegraphics [
height=7.3411cm,
width=9cm
]{ds1705f4.eps}\end{figure} Figure 4: Perturbed wavefront sample generated for r0= 4 cm and $\mathit{L}_{0}$ infinite

  
\begin{figure}
\includegraphics [
height=3cm,
width=14cm
]{ds1705f5.eps}\end{figure} Figure 5: Speckle images obtained from the fractal model. The image field is equal to 64 by 64''2 and the Fried's parameter equal respectively to 2 a), 4 b), 6 c) and 10 cm d)

2.2.2 The simulated point spread function

The psf S of the whole system telescope and atmosphere is given by the square modulus of the Fourier Transform (FT) of the complex amplitude $\Psi$limited to the pupil area:
\begin{eqnarray}
S(\overrightarrow{\mathbf{\theta}})=\left[ FT_{\rm pupil}\left(...
 ...i\left(
\overrightarrow{\mathbf{\xi}}\right) \right) \right] ^{2} \end{eqnarray} (5)
where

\begin{displaymath}
\Psi\left( \overrightarrow{\mathbf{\xi}}\right) =\exp\left( i\varphi\left(
\overrightarrow{\mathbf{\xi}}\right) \right)
.\end{displaymath}

Equation (8) assumed the near field approximation (Roddier 1981). $\varphi$is the turbulent phase generated with the fractal model, $\overrightarrow
{\mathbf{\xi}}$ is the angular frequency vector considered in the pupil plane and $\overrightarrow{\mathbf{\theta}}$ the angular coordinate vector in the focal plane.

$S(\overrightarrow{\mathbf{\theta}})$ corresponds to an image of speckles. In Fig. 5 are represented four typical short exposure speckle images obtained using the procedure developed in the previous section. They are simulated for Fried's parameters r0 equal respectively to 2, 4, 6 and 10 cm and with a pupil aperture D equal to 10 cm. The spatial coherence outer scale of the used randomly perturbed wavefronts has an infinite value. Large boiling and displacements of the speckle centroids are observed for small values of r0 while for r0 equal to the pupil aperture, the speckle image is similar to the diffraction pattern of the instrument.

  
\begin{figure}
\includegraphics [
height=3.5cm,
width=8.5cm
]{ds1705f6.eps}\end{figure} Figure 6: Simulated solar images recorded with a short exposure time and r0 equal to 3 a) and 7 cm b). Solar limbs extracted from images recorded in these seeing conditions (dashed line: r0 = 3 cm and solid line: r0 = 7 cm) c)

We are then able to simulate psf's of the whole system telescope and atmosphere in various seeing conditions and also for different integration times (average of short exposure samples).

2.3 Simulation of recorded solar images in various observation conditions

Solar images as recorded through the terrestrial atmosphere are simulated. Assuming isoplanatism, a solar image is obtained by the convolution of the synthetic image of the Sun (see Sect. 2.1) with the psf of the imaging system (atmosphere and telescope):
\begin{eqnarray}
I(\overrightarrow{\mathbf{\theta}})=O(\overrightarrow{\mathbf{\...
 ...t\vert
\overrightarrow{\mathbf{\theta}}\right\vert \leq\theta_{0} \end{eqnarray} (6)
where $\otimes$ is the convolution symbol and $\theta_{0}$ the isoplanatic angle.

We use this formalism to build solar images recorded in various seeing conditions and with different CCD integration times. The spatial coherence outer scale of the generated perturbed wavefronts is infinite. The pupil aperture D and the observation wavelength are respectively taken equal to 10 cm and $0.55\,\mu{\rm m}$. These kind of images will be useful in the next section to develop the error study on solar diameter measurement. They have similar properties as those recorded with the solar astrolabe (Laclare et al. 1996; Laclare & Merlin 1991).

2.3.1 Solar images recorded in different seeing conditions but with a short exposure time

In this case, solar images are simulated using various Fried's parameters r0 (Fig. 6) but integration times short enough to freeze the turbulent motion of the atmosphere i.e. the integration time t is less than the characteristic evolution time of the atmosphere $\tau$ (Aime et al. 1986). For a given simulated image sequence, Fried' s parameter r0 remains constant but images are obtained with independent psf's.
  
\begin{figure}
\includegraphics [
height=3.5cm,
width=8.5cm
]{ds1705f7.eps}\end{figure} Figure 7: Simulated solar image recorded in same seeing conditions (r0 = 5 cm) but with different exposure times ($t \leq\tau$ a) and $t \gg\tau
$ b)). Solar limbs extracted from images recorded with exposure time respectively lower (solid line) and higher (dashed line) than the atmospheric characteristic time $\tau$ c)

2.3.2 Solar images recorded in same seeing conditions but with different exposure times

Long exposure solar images are obtained by the average over a great number of short exposure frames. The integration time t of the long exposure images is then equal to $n\tau$ where n is the number of averaged images and $\tau$the atmospheric characteristic time. Figure 7 shows the simulated solar images recorded in same seeing conditions ($r_{0} = 5\,{\rm cm}$) but with different integration times.

  
\begin{figure}
\includegraphics [
height=5cm,
width=17cm,
]{ds1705f8.eps}\end{figure} Figure 8: Analysis of an image sequence composed of short exposure frames recorded for a Fried's parameter equal to 3 cm. a) Typical simulated solar limb extracted from a solar image. b) Second derivative of the solar limb function and definition of the inflexion point. c) Reconstructed trajectories of both direct and reflected solar images approximated by straight lines using least-square fits

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