6 Conclusion

We have seen that the use of the model (which can be considered as an interpolating function) gives results in agreement with many other observations studied here, which are also corrected for the disruptive effects. In the same manner, the results obtained by the derivative method not extrapolated agree with the others results not corrected, or with the visual observations.

The fundamental difference between the two methods is a different use of the data. The relevant parameter, which is the abscissa of the derivative extremum, is obtained in two different ways:

1.

By numerical method, an approximate detection is done, followed by a precise numerical computation around the approximate position. Only few data are used in this final computation, around 40 pixels around the first approximate value. On this interval, the abscissa of the derivative extremum is computed by a barycentric method, in which the points where the intensity is greater than half the height of the derivative peak are taken into account. The final result is obtained using the Fried parameter, determined by statistical method, in correlation with others parameters as the width of the derivative and the solar radius. A following extrapolation to the limit value of r₀ gives the final value of the radius;

2.

By the model, a maximum of intensity values along a CCD line are taken into account in order to find the model parameters. But, when the approximate inflection point is too closed to the extremities of the line, the least squares fit program cannot run correctly and the corresponding line is eliminated. Nevertheless, the use of the model gives directly the good value of the solar radius. One can see that the computation of r₀ and of the derivative width is not used to obtain the correct radius but only to evaluate the perturbing effects.

A new remark which can be done, is that, despite the relatively low precision obtained for the true radius determination by the derivative method (mainly due to the extrapolation), it is very satisfactory to obtain practically the same result with the two methods which are, in theirs principles, completely different.

Another remark turns around the extrapolation done by numerical computation. These results agree very well with the model results, despite the hypothesis we have to assume, which are:

• the parameter r₀ represents correctly the disruptive effect of the atmosphere;
• the extrapolation, founded on its infinite value for a perfect atmosphere, is valid.

It was not clear that the r₀ parameter, which is obtained by statistical information about the solar trajectory, could permit to extrapolate the results to $r_0\rightarrow\infty$ and conduct to the same results as with the model. Certainly, the amount and the quality of data obtained during several short campaigns are strong favorable factors. Nevertheless, the use of the model do not need to use statistical data. The correct results are immediately obtained with less error (0$.\!\!^{\prime\prime}$02 in place of 0$.\!\!^{\prime\prime}$08). The second difference is that the model uses, along each CCD line, the maximum possible of data to calculate the five parameters a, b, p, x₀ and c. The parameters a and c cannot be use here as they contain photometric information about the solar intensity and that photometric reference is not observed.

In our opinion, we think that it is better to use the model for several reasons. Firstly, there is a better use of the data along the CCD line, that implies that the determination of the inflection points should be better. Effectively, the least squares fit precision in the parabola fitting - made to determine the reference point, extremity of the vertical radius - is better when the model is used. Secondly, the Fried parameter, determined with $\sigma$ obtained for the reference point trajectories is less affected by the preceding successive calculations, and particularly the one done to find the derivative peak abscissa.

Finally, the quality of the CCD observations allows us to have an evaluation of the disruptive effects existing in the observations and to obtain the correct value of the solar radius, which obeys to the assumed definition. The two reduction methods used give the same result, provided that the numerical calculation of the derivative can be correctly extrapolated. But, due to this extrapolation, the result obtained by this method is more precise. We have also, perhaps, the possibility to correct the old observations. It is what we wish.

In a future work we hope to give new results obtained by observations done at Calern Observatory without rotating shutter. The images are much better that the ones obtained with the shutter and analyzed here. So, we hope to have also better results, and to be able to analyze carefully the observations made in Rio de Janeiro and San Fernando Observatories, with which we have very good cooperation, and the future ones made at Antalya (Turkey) where a new campaign will begin in 1999 with a CCD astrolabe.