3 Determination of the solar radius, following its new definition

  

Figure 1: Derivative methods results: linear fits and extensions for solar radius (R) and derivative width ($W_{\rm d}$) as functions of the Fried parameter (r0). The value $1/r_0=0~{\textrm{ m}}^{-1}$ corresponds to an ideal atmosphere, for which $r_0\rightarrow\infty$. a): solar radius and Fried parameter; b): derivative width and Fried parameter

The adopted method gives very good and homogeneous results but cannot solve easily the problem of the solar limb definition, due to the center to limb darkening. It is well known that the maximum of the derivative along a solar image section define the limb of an apparent Sun, which is not the true limb of the Sun. This effect is the result of the combination of the diffraction, the center to limb darkening and the atmospheric turbulence, which have as result to shift systematically the inflection points towards the center of the Sun image. Consequently, the apparent solar radius is always smaller than the true one ([Rösch et al. 1996]).

Considering the amount of information obtained during each transit, it seems possible to evaluate this effect and to derive the corresponding corrections to be applied to the results.

The ways to suppress the preceding effects seem to be:

• to extrapolate the results of the numerical analysis by a method which needs to be defined, or
• to construct a model of the solar images, which takes account of all the physical effects.

Only the CCD observations done with the rotating shutter will be considered here. After a precise description of the adopted model, an application to solar measurements made at Calern Observatory will conduct to new corrected results.

The results obtained by numerical derivation of the same data are analyzed and extrapolated. These results allow us to compare the two methods.

3.1 The numerical derivation and its extension

Before the construction of the model, the problem led by the center to limb darkening and the atmosphere was known and numerical studies were done to suppress it. With the first reduction method, it was a priori impossible to evaluate directly the necessary corrections. Nevertheless, correlations have been established between the observed radius and other parameters as the Fried parameter given by:

$$r_0=8.25\ 10^5\cdot D^{^{-\frac{1}{5}}}\cdot\lambda^{^{\frac{6}{5}}}\cdot(\sigma^2)^{^{-\frac{3}{5}}}$$

which may be considered as a representation of the atmospheric turbulence ([Irbah et al. 1994]). D is the aperture of the astrolabe refractor, $\lambda$ is the wavelength ($\mu$m) and $\sigma$ is the standard deviation of the linear fit on the observed trajectories.

This formula giving r₀, which depends on $\sigma$, shows that the quality of the sky is correlated with r₀, the best sky corresponding to $r_0\rightarrow\infty$ or to a null value for $\sigma$.

Another interesting parameter is the derivative width, defined at half height of the derivative peak, or, also, as the distance between the two inflection points of the derivative. We use here the daily mean value of each quantity (Table 1).

The first study shows that the Fried parameter and the width of the numerical derivative are strongly correlated (for more details, see [Sinceac 1998]; we use here these results only to validate the new method). If r₀ is the Fried parameter and $W_{\rm d}$ the width of the derivative obtained numerically, we find that (Fig. 1b):

$$W_{\rm d}=0\hbox{$.\!\!^{\prime\prime}$}^{{\rm .m}}07245\cdo... ...4.^{{\rm m}^{-1}}1188\right)+7\hbox{$.\!\!^{\prime\prime}$}137.$$

Excepted the conventional mean value of $\frac{1}{r_0}$, $24.1188 ~{\rm {m}}^{-1}$, the numerical coefficients are respectively known with the standard errors $\pm 0\hbox{$.\!\!^{\prime\prime}$}^{\rm {.m}}00416$ and $\pm 0\hbox{$.\!\!^{\prime\prime}$}022$ for the constant term.

In the same manner, the correlation between r₀ and the observed solar radius R was computed (Fig. 1a):

$$R=-0\hbox{$.\!\!^{\prime\prime}$}^{\rm {.m}}00719\cdot\left(... ...{{\rm {m}}^{-1}}1188\right)+959\hbox{$.\!\!^{\prime\prime}$}455$$

with the respective errors on the constants: $\pm 0\hbox{$.\!\!^{\prime\prime}$}^{\rm .m}00307$and $\pm 0\hbox{$.\!\!^{\prime\prime}$}013$.

The Fried parameter represents the atmosphere effects that we want to remove. As we know that the ideal atmosphere corresponds to an infinite value for r₀, it is interesting to examine which the values of $W_{\rm d}$ and R are when the Fried parameter is extrapolated to $+\infty$. For $\frac{1}{r_0}=0$, we obtain:

$$R = 959 \hbox{$.\!\!^{\prime\prime}$}628 \pm 0 \hbox{$.\!\!^{\prime\prime}$}075.$$

This is practically the value given one century ago by [Auwers 1891]! The derivative width becomes equal to:

$$W_{\rm d}= 5\hbox{$.\!\!^{\prime\prime}$}39 \pm 0\hbox{$.\!\!^{\prime\prime}$}10.$$

These results may represent the values, obtained with the same instrument, of the solar radius and of the derivative width along the CCD lines, through a perfect atmosphere or above the atmosphere. Due to the extrapolation method, one can see that the precision is not so good.

3.2 The model

We have shown that the phenomena conducting to an apparent solar image are, mainly:

• a center to limb darkening of the solar intensity, including absorption and turbulence coming from the terrestrial atmosphere;
• the instrumental diffraction effect.

It is well known that the perfect diffraction curve of an uniform extended source through a circular aperture presents a maximum of its derivative exactly on the true position of the limb of the source surface ([Chrétien 1959]). If the source intensity is not constant, the position of this point changes and generally does not coincide with the true limb.

We will now consider a CCD line section of the apparent image. Without darkening effect, the representative intensity curve is assumed to be a hyperbolic tangent curve as $\tanh(x)+1$ (the value 1 is added to obtain always positive values). The resulting function (of the form $\frac{2\mbox{e}^x}{\mbox{e}^x+\mbox{e}^{-x}}$) is multiplied by a first unknown constant a, in order to represent the signal amplitude. The abscissa of the inflection point is obtained for x=0, and a change of a parameter is used to put it in x₀. By the same way, the slope of the intensity function is assumed to be different from 1 and depends on a constant b. So the x parameter is changed to $b\cdot(x-x_0)$.

Finally, an unknown constant c is added to give account of the added intensity due to noise and sky lighting.

With these assumptions, the intensity model along a CCD line, due to diffraction effect only, can be written as:

 

$$I_0(x) = c + 2 \cdot a\cdot\frac{\mbox{e}^{b(x-x_0)}}{\mbox{e}^{b(x-x_0)}+ \mbox{e}^{-b(x-x_0)}}\cdot$$ (1)

In Eq. (1), x₀ is the abscissa of the point of the solar limb which represents the diameter extremity verifying the adopted definition as this function represents a perfect extended source with constant intensity. In order to give account of the center to limb darkening, we have to adopt a mathematical representation. Here a simple model of this effect is adopted, which has exactly the same form as the perfect model, but with a slope parameter p, which has always the same sign as b and is much lower. The final mathematical form of the model consists to multiply the part of the perfect model which contains a by the center to limb darkening model.

The result, more realistic, can be written now:

 

$$I(x)=c+4a \frac{\mbox{e}^{b(x-x_0)}}{\mbox{e}^{b(x-x_0)}+\mb... ...{\mbox{e}^{p(x-x_0)}}{\mbox{e}^{p(x-x_0)}+\mbox{e}^{-p(x-x_0)}}$$ (2)

One can remark that the perfect model (Eq. 1) and the one for center to limb darkening have the same position for the inflection point x₀. This fact has no physical significance and if we adopt a different position for the two inflection points, nothing change in the results, except the complexity of the theoretical and practical calculations. As this model, we will see, is convenient, we do not keep this supplementary unknown.

  

Figure 2: Section along a CCD line: least squares fitting of the model on real data (the vertical line marks the abscissa of the inflection point, x0). The vertical scale gives the intensity level of the signal in a scale defined by the CCD camera (intensity levels between 0 and 255)

A second remark may be that far from the limb, these functions do not give an exact representation of the image, and that they can be used only in the vicinity of the limb. In fact, the intensity I(x) increases from c to c+4a, which is not the case for the solar light intensity, which increases from a minimum to a maximum and decreases to the minimum. In fact, the entire solar image may be represented by the product of a by two functions of b, p and $x^{\prime}_0$ for the first one, and of -b, -p and $x^{\prime\prime}_0$ for the second one (Eq. 4). In this case, in the vicinity of, saying $x^{\prime}_0$, the functions which contain $x^{\prime\prime}_0$ are practically equal to 1, and vice versa. As, on the CCD, the field is very little and contains only one of the two possible inflection points, we are always in the preceding situation and, consequently, we will use the simple Eqs. (2) or (3), by changing the b and p signs following the observed part of the Sun.

We give, in the Appendix, all the necessary expressions needed to use this model. The notations adopted in Appendix will be used in the following.

Now, it is necessary to know where is the abscissa of the zero of the second derivative of the perfect and realistic model. Starting with the Eq. (1), the zero of the second derivative is given by the equation $\frac{{\rm d}^2U}{{\rm d}x^2}=0$ which is equivalent (see Appendix) to:

$$X-\frac{1}{X}=0$$

This function is equal to zero if X²=1 or x=x₀ following the expression of X. The ideal function, without darkening, reaches its extremum in x₀.

Now, the final model, given by Eqs. (2) or (3) has to be derived in order to find the abscissa of the inflection point. The second derivative of this function is given by the Eqs. (5) or (6).

The terms containing second derivatives are equal to zero for x=x₀, but the term $\frac{{\rm d}U}{{\rm d}x}\cdot\frac{{\rm d}V}{{\rm d}x}$ is different from zero.

We can conclude now that the inflection point abscissa is not the same in the Eqs. (1) and (2). As the interesting parameter is x₀, one way to get it is to solve the Eq. (6) for which the solution is function of the parameters x₀, b and p only. This solution $x_{\rm s}$ will contain the difference with the true value x₀ and solve the problem, provided that b and p are known. It seems better to fit, by least squares method, the model on the data obtained along a line. This method gives directly x₀ and the others parameters. Figure 2 shows a section on a CCD line and the model obtained by this method. Anyway, an approximate solution of the Eq. (6) was calculated to evaluate the difference $x_{\rm s}-x_0$, assuming that this difference is little as b and p are small and as, normally, $\vert p\vert<\frac{\vert b\vert}{10}$. The calculation, made at the first order of the parameters (see Appendix for more precise formula), gives:

$$x_{\rm s}-x_0=\frac{p}{b^2}\qquad \mbox{or}\qquad x_{\rm s}=x_0+\frac{p}{b^2}\cdot$$

In the model, b and p have always the same sign. If p is negative, the center of the Sun is in the direction of the negative x and $x_{\rm s}<x_0$, and the contrary if p>0. This shows that, as it was write previously, the apparent diameter of the Sun is always smaller than the true one. With the usual values of $b\simeq 0.15$ and $p\simeq 0.01$ the difference between real and apparent radius is approximately equal to 0.44 pixel or $\simeq 0\hbox{$.\!\!^{\prime\prime}$}33$.

Considering the high value of this effect, it seems much more convenient to determine the model parameters, by fitting it to the data and using the whole useful data along the CCD line (which turns around 80 kept values for the intensities). The main result is that, effectively, the new obtained radius are greater than the previous ones obtained by numerical research of the first derivative extremum.

  

Table 1: 1996-1997 CCD measurements campaign, daily mean results