2. THE STATISTICAL RATE OF STELLAR OCCULTATION BY KBOs:
A FIRST APPROACH

The occultation statistical rate is the probable number of occultations of a star detected during a given time interval. When an observer (on the Earth in most cases) crosses the shadow of a KBO cast by a star, the star light is not fully extinguished and the stellar flux oscillates: If these oscillations are detected, there is an occultation, and the whole region where these oscillations can be detected is called the diffraction shadow. So, the occultation rate is computed as the probability of intersection in the sky plane of the star disc with the KBOs diffraction shadows.

If, at first, we ignore diffraction, the definition of an occultation event simplifies: there is an occultation when the observer crosses the geometric shadow of a KBO, and the star light is fully or partially extinguished, depending only on geometric parameters linked to the KBO and to the star. In addition, the detection of an occultation event depends on the conditions of the observation and in particular on the instrument used for recording the occultation. In the following discussion, we estimate this ``geometric'' occultation rate by KBOs for a given star; we defer the possibility of modeling and exploiting the diffraction phenomenon to later sections.

We consider a population of KBOs moving with respect to a star in the sky plane. There is an occultation of the star by a KBO if the minimum distance between the two objects is smaller than the sum of their radii. We may then write the number of geometric occultations of a star with angular radius $\Phi _\star$ and with an inclination i on the Ecliptic plane, by KBOs of angular radius between $\varphi$ and $\varphi + d\varphi$, observed during an interval $\Delta t$ as:

$$n_{\rm {occ}}(\varphi)= 2 \delta(\varphi,i) (\varphi+\Phi_\star) v_o \Delta t$$ (1)

where $\delta(\varphi,i)$ is the density of KBOs in the sky plane and $v_o$ is the KBOs velocity in the sky plane.

The total number of occultations during $\Delta t$ is then:

$$N_{\rm {occ}} = \int^{\varphi_{\rm {max}}}_{\varphi_{\rm {lim}}}n_{\rm {occ}}(\varphi)d\varphi$$ (2)

where $\varphi_{\rm {max}}$ is the maximum expected angular radius of KBOs (this value is not critical since we know that big KBOs are very rare, i.e. $\delta(\varphi,i)$ must be vanishingly small for large $\varphi$) and $\varphi_{\rm {lim}}$ is the angular radius of the smallest KBO detectable by occultation.

Let us now discuss the three main parameters governing this geometric occultation rate, which are:

1. The density of the KBOs which is poorly known: The high magnitude of these objects limits the size of the directly detectable objects to few kilometers, but we assume that the Kuiper Belt has a size distribution $\propto \rho^{-q}$ with a constant index q extending to meter-sized objects. The Near Earth Asteroid (NEA) population (the only small body population in the Solar system known down to meter-sized objects) indeed has a size distribution with an index more or less constant to 5m radius [Rabinowitz et al. 1994]. We will return to the assumption of a constant index q in section 4. In addition, the KBOs are assumed to follow an exponential distribution in inclination with an angular scale height H and the spatial density of KBOs is also assumed to depend on the distance to the sun, D , with the same law as the protoplanetary disk, i.e. as $D^{-2}$. If we assume N KBOs larger than 1km exist, located between 30 and $D_{\rm {max}}$ AU, the previous assumptions lead to the following 2-D differential size distribution $\nu$, which is the number of KBOs of radius between $\rho$ (in km) and $\rho+d\rho$, with inclination between i (in degrees) and $i+di$, and at an heliocentric distance between D (in AU and >30 AU) and $D+dD$:
   

   $$\nu(\rho,i,D) = \frac{180 N}{4\pi^2}~ \frac{D^{-2}}{(D_{\rm {max}}-30)}~ \frac{C(H)}{H}e^{\frac{-i}{H}}~ (q-1)\rho^{-q}$$ (3)

   
   
   with $C(H)= \frac{\pi[(180/\pi)^{2} + H^{2}]}{180[180/\pi + H e^{-90/H}]} \simeq 1+\sin^{2}H$, where H is expressed in degrees, and $\nu$ is given per AU and per steradian.

   Recent observations allow us to estimate $10^{11}$ objects larger than 1 km, located between 30 and 50 AU (the practical limit of existing surveys), a differential size distribution index $q\approx 4$ and a latitudinal scale height $H=30\hbox{$^\circ$}$ [Jewitt 1999]. Then, by substituting these values in Eq.3 and by integrating in D between 30 and 50 AU, we obtain the following mean density of KBOs in the sky plane of angular radius between $\varphi$ and $\varphi + d\varphi$ and with inclination between i and $i+di$:
   

   $$\delta(\varphi,i)=3~10^{-11} e^{\frac{-i}{30}} \varphi^{-4}$$ (4)

   
   
   where, for convenience, $\varphi$ is expressed in mas(= $0.001^{\prime\prime}$) and $\delta$ is so given per $\rm {mas}^{2}$. Let us note that, if we assume the KBOs size distribution ranges from one meter to some hundreds of kilometers (that is from about $3.10^{-5}$ to $10\rm {mas}$ at 40 AU), we can estimate an optical depth of the Kuiper Belt in the ecliptic of $\tau_{\rm {KB}} \approx\int_{3.10^{-5}}^{10} \delta(\varphi,0) \varphi^2 d\varphi \approx 10^{-6}$. We will show however in section 4 that diffraction perturbs the stars light far away from the geometric disc, so that the proportion of the ecliptic sky perturbed by the Kuiper Belt is much larger than that inferred by the optical depth $\tau_{\rm {KB}}$.

2. The velocity of the KBOs in the sky plane $v_o$ which is involved in both the number of occultations ( $N_{\rm {occ}}\propto v_o$) and the duration of the occultation ( $dt_o \propto 1/v_o$). It is given as
   

   $$v_o = v_E \left( \cos\omega -\sqrt{\frac{1}{D_{\rm {AU}}}} \right)$$ (5)

   
   
   where $D_{\rm {AU}}$ is the heliocentric distance of the KBOs in astronomical units, $v_E$ is the Earth's orbital speed, $\omega$ is the angle between the KBO and the anti-solar direction which is called "opposition" hereinafter.

   Because $v_o$ depends on the distance D of the occulting objects, so do the probability and the duration of the occultations. This property could allow the discrimination between an occultation by a KBO and an occultation by an asteroid or a comet (see section 6). By fixing D to 40 AU and having $v_E\approx 30$km/s, we get $v_o \approx 30(\cos\omega-0.16)~\rm {km/s}$ ( $\approx (\cos\omega-0.16)$  mas/s). Then $v_o$ is maximum ($\approx$ 25 km/s) toward the opposition and decreases toward a direction, called the "quadrature" hereinafter, for which $\cos \omega \simeq 1/\sqrt{D_{AU}}$, i.e. $\pm 81\hbox{$^\circ$}$ for an object at 40 AU. An occultation, usually very brief, becomes slower in this direction. For instance, if an occultation by a 1km radius KBO was detected, it could last from about 0.1 to few seconds, as $\omega$ goes from the opposition to the quadrature.

3. The size of the smallest KBOs detectable by occultation $\varphi_{\rm {lim}}$ depends on the angular radius of the star $\Phi _\star$ and on that of the KBO itself: An object which is too small passing in front of the star will not generate a detectable decrease in the stellar flux. For a KBO smaller than the stellar disc, geometrical optics gives the straightforward condition:
   

   $$(\frac{\varphi}{\Phi_\star})^2 \geq \epsilon$$ (6)

   
   
   where $\epsilon$ is the detection threshold or sensitivity of the instrument. This yields for instance a detection threshold $\varphi_{\rm {lim}}=2.\Phi_\star\sqrt{\sigma}$ at $\epsilon=4\sigma$, where $\sigma$ is the r.m.s. signal fluctuations (i.e. the photometric precision limit of the instrument), and with $\Phi _\star$ typically about 0.01 to 1 mas. But, as we shall see in subsequent sections, these formulae are quite approximate and only valid for a large stellar radius (>0.1 mas) because diffraction is not taken into account: the real detection threshold not only depends on the star size in a very different way than stated here, but also depends on both the distance from the KBO to the observer and the observation wavelength.

We may finally deduce from Eq.2 and Eq.4 and the discussion above, the following approximation of the total number of geometric occultations by KBOs of a star with an angular radius $\Phi _\star$ and an inclination i on the Ecliptic:

$$N_{\rm {occ}}(\Phi_\star,i) \approx 2~10^{-7} {v_o} {\Delta ... ...m}}}^{10\rm {mas}}{\varphi^{-4} (\varphi+\Phi_\star) d\varphi}$$ (7)

where $v_o$ is expressed in mas/s, ${\Delta t}$ in hours, $\varphi_{\rm {lim}}$ and $\Phi _\star$ in mas. For instance, we may estimate the geometric occultation rate per night (8 hours) in the Ecliptic for a $\epsilon=4\sigma$ sensitivity, and $\Phi_\star > 0.1$ mas as:

$$N_{\rm {occ}}(\Phi_\star,0) \approx 1.5~10^{-7} {v_o} \Phi_\star^{-2} \sigma^{-1.4}$$ (8)

So this geometric occultation rate per nigh is less than 0.1 in the most favourable case, with a typical KBO detection threshold of a few kilometers in size. Fortunately, as we will show in section 4, this result, which does not take into account any diffraction effects, significantly underestimates $N_{\rm {occ}}$ .