4 Results

In this observational mode, where the two images are separated, each CCD frame contains only the direct or the reflected image of the Sun. The acquired data are contained in a window of 256 lines$\times$101 columns. A first least squares fitting along each line uses only 101 values of the intensities to determine the parameters x₀, a, b, p and c of the model.

The calculation of the observed radius follows the same method used after numerical determination of the inflection points ([Sinceac 1998]; [Sinceac et al. 1998a]).

The Table 1 gives all the results obtained by observations done with the rotating shutter. They are given for the first method of reduction (position of the extremum of the first derivative) and for the method using the model. In each image, the fact that the CCD window (only 101 columns) is not centered on the solar image doesn't affect the results for the derivative method, which uses practically all the CCD lines (the intensity function is derived only in the vicinity of the inflection point, [Sinceac 1998]). To find the parameters of the model by the second method, the software needs more informations that the first one and more CCD lines will be eliminated, but the model detects better the problems. To be able to compare the two set of results, the two analysis modes use approximately the same data. In fact, the elimination procedure of bad CCD images being not the same, the set of data used for each method cannot be strictly identical (there is a very great quantity of data). Finally, it is the method of least squares used for the parabolas on each image which decides possible removal of lines and/or images ([Sinceac 1998]). This explain why the results by derivative method given here are not exactly the same as the ones published in our precedent paper ([Sinceac et al. 1998a]) and on the Table 1. The mean values are:

The dispersion $\sigma$ of the results is practically the same with the two methods and equal to 0$.\!\!^{\prime\prime}$14.

Using the extrapolation for $r_0\rightarrow\infty$ in the derivative method, we have:

$$\quad\quad R = 959\hbox{$.\!\!^{\prime\prime}$}63 \pm 0\hbox{$.\!\!^{\prime\prime}$}08$$

which is not so precise.

The difference found between the radius obtained by each method shows that the values of the constants b and p are not far from the ones given in the preceding paragraph. In fact, during the calculations, the values encountered for b and p turn respectively around 0.15 and 0.01 pixel^-1. But few variations appear, due mainly to the atmosphere and the integration time of each image (0.02 second).

It is interesting to compare these values with the theoretical results obtained using the model. We have just seen that the mean solar radius, obtained with the derivative method and extrapolated for $r_0\rightarrow\infty$, becomes practically the same as the one obtained with the model. The successive correlations calculated between the correction to the apparent inflection point $x_{\rm s}$ (Appendix) and the new computed r₀ give, in the same units:

$$x_{\rm s}-x_0=\frac{p}{b^2}=0.01394\cdot\left(\frac{1}{r_0}-23.4369\right)+0.4322$$

with the respective errors $\pm 0.00055$ and $\pm 0.0032$.

The new derivative width is also correlated with r₀ and we have:

$$W_{\rm m}=0.1798\cdot\left(\frac{1}{r_0}-23.4369\right)+9.272$$

with the respective errors: $\pm 0.0066$ and $\pm 0.039$. The calculation for $\frac{1}{r_0}=0$ gives:

$$\frac{p}{b^2}=0.105\pm 0.013\qquad\mbox{and}\qquad W_{\rm m}=5\hbox{$.\!\!^{\prime\prime}$}06\pm 0.04.$$

The successive model derivatives show that the theoretical derivative width $w_{\rm h}$ is equal to $\frac{1.8708}{b}$, in pixel (Appendix). The preceding result being identified with $w_{\rm h}$, it becomes possible to calculate b from the theoretical value of $w_{\rm h}$. As the pixel value is $0\hbox{$.\!\!^{\prime\prime}$}74$, we obtain (in pixel^-1):

$$b=0.2736\pm 0.0028.$$

Immediately, the value of $\frac{p}{b^2}$ give us the one of p (in pixel^-1):

$$p=0.0079\pm 0.0016.$$

  

Table 2: Some recent results for the apparent solar radius, obtained by different methods and instruments (annual and/or general mean values)

The obtained value for b is much greater than the mean value calculated with the data of the Table 1. If, by the model, we research the maximum value of the slope of the curve representing the intensities along a CCD line, we obtain $a\cdot(b+p)$. The obtained high value for b may be interpreted as for a perfect or outside the atmosphere, where, logically, this parameter is much greater than the one detected through the real atmosphere.

For the parameter p, the result is opposite: the value outside the atmosphere is less than the observed one. This result is also logical as the center to limb darkening effect can be essentially increased by the transmission of the terrestrial (and solar?) atmosphere and not so much by its turbulence.

Considering the relatively little amount of data, the information obtained here are sufficiently precise to find coherent results, and give very interesting first information about the disruptive effects introduced by the atmosphere. The Table 1 shows also that the parameters b and p are finally relatively stable since the b values stand between 0.129 and 0.164 and the p values between 0.0090 and 0.0103.

We can consider now that $b=0.274\pm 0.003$ is, outside the atmosphere, the value of the main parameter defining the solar light intensity slope near the inflection point of the limb. In the same conditions, the parameter $p=0.0079\pm 0.0016$ represents the slope parameter of the center to limb effect, near the limb. Finally, the solar radius, corrected for the atmospheric and darkening effects is equal to: