3.1 Basics

The light emitted by a point source (assumed to be at the infinity to yield planar waves) incident on a sharp-edged obstacle (such as a KBO) is diffracted. Owing to the Huygens-Fresnel principle of wave propagation, each point of a wave front may be considered as the center of a secondary disturbance giving rise to spherical wavelets, which mutually interfere. If part of the original wave front is blocked by an obstacle, the system of secondary waves is incomplete, so that diffraction occurs. When observed at a finite distance D from the obstacle, this effect is known as ``Fresnel diffraction'', and falls within the scope of the Kirchhoff diffraction theory which remains valid as long as the dimensions of the diffracting obstacles are large compared to the observed wavelength $\lambda$ and small compared to D [Born and Wolf 1980, cf.]. The characteristic scale of the Fresnel diffraction effect (that is, roughly speaking, the broadening of the object shadow) is the so-called Fresnel scale $\sqrt {\lambda D/2}$. The Fresnel scale at 40 AU, observed at $\lambda = 0.4 \mu$m, is 1.1 km, i.e. a KBO typical size, and so the diffraction must be seriously taken into account to analyze the occultations by KBOs.

Let us now consider the case of a monochromatic point source occulted by an opaque spherical object of radius $\rho$. If $r$ denotes the distance between the line of sight (the star's direction) and the center of the object, and if the lengths $\rho$ and $r$ are expressed in the Fresnel scale unit (noted Fsu hereinafter), the normalized light intensity $I_\rho (r)$ is given by (see appendix B of [Roques, Moncuquet and Sicardy 1987])
Outside the geometric shadow ($r \ge \rho$) :

$$I_\rho (r) = 1 + U_1^2(\rho,r) + U_2^2(\rho,r) - 2 U_1(\rh... ...)} \protect + 2 U_2(\rho,r) \cos { \frac{\pi}{2}(r^2+\rho^2)}$$ (9)

Inside the geometric shadow ($r \le \rho$) :

$$I_\rho (r) = U_0^2(r,\rho) + U_1^2(r,\rho)\protect$$ (10)

where $U_0,U_1$ and $U_2$ are the Lommel functions defined by (for $x \le y$)

$$U_n(x,y) = \sum_{k=0}^\infty { -1^k (x/y)^{n+2k} J_{n+2k}(\pi xy)}$$ (11)

where $J_n$ is the Bessel function of order n.

Fig.1(top) shows the diffraction pattern of a circular object (1km radius) occulting a point star and is computed using Eq.9 and Eq.10. The Fresnel scale is set to 1km. The size of the shadow is larger (about one Fresnel scale) than the geometric shadow, and overall the diffraction fringes are visible at a large distance from the object (as long as the photometric sensitivity is good).

Figure 1: The shadow pattern of a 1 km radius KBO in front of a point star. The two horizontal axis are distances in kilometers and the vertical axis is the normalized stellar flux. Top:The KBO is circular. Bottom : The KBO is irregular (the limb is $\sim 6\%$ corrugated) and elliptical (the eccentricity is 0.7 and the half major axis is 4/3 km, such that $2/3<\rho <4/3$ km). The projected contours show the $\pm 4\%$ variation of the light intensity (the black filled areas correspond to $+4\%$, i.e. $I>1.04$). The grey central spot indicates the exact geometric shadow of the KBO.