3.3 What could be finally detected thanks to diffraction ?

From the previous discussion, it appears that the diffraction effect during a stellar occultation by a KBO (and more generally by any small object in the Solar System) will depend as usual on its size and on the apparent distance from the star to the object (the impact parameter), but also strongly on the apparent stellar radius and on the Fresnel scale -that means (i) that the star has to be well selected to optimize its apparent size and flux at the observed wavelength $\lambda$, and (ii) that such observations will generally need another means of estimating the distance D of the occulting object we are dealing with. We will return later to these two points, but, if we assume these observational parameters have been fixed, the ability to detect or not detect an object of a given size will then only depend on the photometric precision of the instrument.

To show this dependence in the most general way possible, the figures 3a to 3e gives the iso-levels of fluctuation, that is the maximum amplitude of the fluctuation of the stellar intensity, as a function of the real radius of the object and of the distance from the star to its center, both expressed in Fsu (and both on logarithmic scale). Each figure corresponds to a given apparent stellar radius (i.e. when projected in the occulting object plane) $R_\star$, which is 1/10, 1/2, 1, 2 and 10 Fsu, respectively. On each figure, the colored zones correspond to the geometric shadow: the blue zone is the complete occultation zone and the yellow one is the partial occultation zone.

Figure 3a: Isocontours of the maximum amplitude variation of the normalized flux of a star occulted by an object, as a function of the distance from the line of sight to the KBO center (horizontal axis) and of the real object radius (vertical axis). The four non-labeled thin contours are the $\epsilon$-isolevels for $\epsilon = 4 \times 10^{-4, -3, -2, -1}$. The grey zones correspond to the solution from geometric optics: the blue zone corresponds to a complete occultation and the yellow one corresponds to a partial occultation. This figure was computed with a ``small sized'' star, that is with an apparent star radius of $1/10 \sqrt {\lambda D/2}$.

Figure 3b: As in Fig.3a, but with an apparent stellar radius of $1/2 \sqrt {\lambda D/2}$.

Figure 3c: As in Fig.3a, but with an apparent stellar radius of $\sqrt {\lambda D/2}$.

Figure 3d: As in Fig.3a, but with an apparent stellar radius of $2 \sqrt {\lambda D/2}$.

Figure 3e: As in Fig.3a, but with an apparent stellar radius of $10 \sqrt {\lambda D/2}$.

Using these Fig.3a-e, we may now define the diffraction shadow for a given sensitivity $\epsilon$ as the region where the diffraction fringes can be detected: $\rho_\epsilon(\rho,R_\star)$ will denote the radius of the diffraction shadow of a given object (called ``diffraction radius'' hereinafter), which can be seen as the ``occultation effective radius'' of the object for a given star and a given photometric sensitivity. That is, on each figure, the isocontour of level $\epsilon$ simply yields the diffraction radius $\rho_\epsilon$ as a function of the real radius object $\rho$. Finally, let us note that the Fig.3a-3e also yield the smallest detectable object $\rho _{\rm {lim}}$ for a given star radius $R_\star$ and a given sensitivity $\epsilon$: it is the minimum reached by an isocontour of level $\epsilon$.

Two points must be outlined from Fig.3a-3e since they will have important consequences on the KBOs occultation rate computation, and more generally on photometric observations of stars in the ecliptic:

1. The diffraction shadow is generally much larger than the geometric shadow, and this ``shadow broadening'' is more pronounced as the apparent stellar disk size decreases and as the photometric sensitivity increases. Moreover, one can see in figures 3a-e that the broadening due to diffraction is about a factor 1000 around or slightly below the $\epsilon = 10^{-4}$ isocontour: that means, by comparison with the $10^{-6}$ KB optical depth estimated in section 2 (assuming a density given by Eq.4), that all the stars of the Ecliptic could have their light perturbed at an amplitude level within the range $\sim [10^{-5},10^{-4}]$. Although such a fluctuation level is of the order of those due to star seismic pulsations, it should not perturb the seismological observations: each occultation event is brief (typically less than one second), so that it involves frequencies $\;\rlap{\lower 2.5pt \hbox{$\sim$}}\raise 1.5pt\hbox{$>$}\;$1Hz, compared to frzquencies of a mHz and below typical of asteroseismology.

2. The apparent stellar radius, $R_\star$, is a critical parameter, essentially because the occultation lightcurve is smoothed on the stellar disk, and this smoothing effect drastically increases the size of the smallest detectable objects (see Table I), while it generally reduces the diffraction radius: For a 1 Fsu object radius and a $10^{-3}$ photometric sensitivity, $\rho_{10^{-3}}\simeq 18$ Fsu for a quasi-point star, $\rho_{10^{-3}}\simeq 7$ Fsu for a 1 Fsu star apparent radius, and $\rho_{10^{-3}}\simeq 10$ Fsu for a 10 Fsu star, since in this case, the light fluctuation reduces to a decrease of $(\rho/R_\star)^2$ (as in Eq.6) due to the simple transit of the object over the stellar disk (that is the geometric partial occultation of the star).