2 The observations methodology

The astrolabe was elaborated in order to give two images of the same star which follow two symmetrical trajectories relatively to an horizontal line. The measurement principle is to find the instant when the two images of the star define a horizontal line. At this instant, the zenith distance of the star is exactly the one defined by a prism, equivalent to a double mirror, associated with a mercury surface. Finally, to find the transit time, the observational principle is to reconstruct the apparent zenith distance variations of each image as functions of time.

In the case of solar observations, we should find the reference points of two images of the same part of the solar limb. These points are the ones for which the tangential lines to the curves defined by the two images of the solar limb are horizontal. If we suppose that these points are known, the observation follows the same way as for the stars.

This principle is valid as well as for visual and for CCD observations. As the astrolabe shows two solar images during the transit, these images are superposed during half of the transit duration. In order to remove this effect, the observations were done, firstly, using a rotating shutter to acquire, alternatively, the direct image and the reflected one on the mercury.

Each image gives the distribution of the apparent solar intensity I(x,y) relatively to the CCD frame. This frame is defined by the CCD lines (parallel to the x-axis) and columns (parallel to the y-axis). The CCD lines are vertical, due to the choice of the best resolution. The subsequent analysis is done in order to obtain the successive positions of the extremity of the vertical solar diameter (VSR) in the same CCD reference frame.

The sets of these positions, one for each successive direct or reflected image obtained during the limb transit, show the trajectories of the extremity of the VSR seen directly or reflected on mercury. These coordinates are functions of time $x_{\rm d}(t)$ and $y_{\rm d}(t)$ (direct images), and $x_{\rm r}(t)$ and $y_{\rm r}(t)$(reflected images) relatively to the CCD frame. The knowledge of these functions gives us the possibility to determine:

• the transit time of the observed VSR;
• the correction on the CCD line inclination, and consequently the corresponding correction on the observed transit time;
• the pixel sizes on the sky (in arcseconds);
• the true extremity of the VSR along the limb.

The transit time is obtained simply when the two coordinates $y_{\rm d}(t)$ of the direct image and $y_{\rm r}(t)$ of the reflected one are equal. The coordinates $x_{\rm d}(t)$ and $x_{\rm r}(t)$ are not well determined but are useful to calculate, with a sufficient precision, the previous corrections.

Finally the comparison of the transit times obtained for each edge of the Sun, allows us to obtain the observed correction to the apparent solar diameter . Simple subsequent calculation gives the corrected apparent radius for the unit distance (1 AU).