Measurement Technique

In theory, the fluctuations of the normal logarithm of the light intensity is the quantity to compute. So, if $x$ and $y$ are the intensities in two apertures, the normal scintillation index $s_x$, $s_y$ and the differential index $s_{xy}$ should be computed as :

$$s_x=\left\langle\big(\log x-\overline{\log x}\big)^2\right\rangle$$

$$s_y=\left\langle\big(\log y-\overline{\log y}\big)^2\right\rangle$$ (5.1)

and

$$s_{xy}=\left\langle\big(\log(x/y)-\overline{\log(x/y)}\big)^2\right\rangle$$ (5.2)

In fact, the calculations are done by different formulae that replace logarithms with ratios. These "linear" formulae are better suited for the subtraction of photon noise because the latter can be evaluated theoretically. The photon noise can be quite large, so the use of logarithmic formulae for index calculation seems problematic. The linear one are :

$$s_x=\left\langle\big(x/\overline{x}\big)^2\right\rangle-1$$

$$s_y=\left\langle\big(y/\overline{y}\big)^2\right\rangle-1$$ (5.3)

and

$$s_{xy}=\left\langle\big(x/\overline{x}-y/\overline{y}\big)^2\right\rangle$$ (5.4)

The above equations are used in the reduction software to calculate the various scintilation indexes. For the measurments of the intesity of light in the two apertures, packages from Starlink are used such as Photom and Extractror. The reduction code is in the addendum.