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Astron. Astrophys. 343, 916-922 (1999)

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2. A model of cometary occultation

As pointed out by Lamers et al.(1997), a geometric distribution of the dust must be assumed in order to evaluate the photometric stellar variation due to extra-solar comets. Then, by taking into account the optical properties of the cometary grains, the photometric variation can be estimated.

2.1. Distribution of dust in a cometary tail

The evaluation of the distribution of the dust in a cometary tail can be made through particle simulation. The input parameters are the comet orbit, the dust production rate, the ejection velocity and the size distribution. We assume a size distribution [FORMULA] of the form

[EQUATION]

as observed in the solar system, where s is the dust size. We take [FORMULA]m, [FORMULA], [FORMULA], and [FORMULA]m (Hanner 1983, Newburn & Spinrad 1985). This distribution starts at [FORMULA]m with [FORMULA], peaks at [FORMULA]m, and has a tail similar to a [FORMULA] distribution for large sizes.

Dust sensitivity to radiation pressure is given by the [FORMULA] ratio of the radiation force to the gravitational force. We take

[EQUATION]

where [FORMULA] is the luminosity-mass ratio of the star. So the orbits of the small grains are more affected by radiation pressure than those of large grains. This is a very good approximation for particles larger than [FORMULA]m of any realistic composition and for the solar spectrum (Burns et al. 1979).

The dust production rate P is assumed to vary with r, the distance to the star, and is taken to be

[EQUATION]

(see, for example, A'Hearn et al., 1995; Weaver et al., 1997, Schleicher et al., 1998). The dust production is taken to be zero beyond 3 AU for a solar luminosity.

As soon as a grain is produced from a comet nucleus, we assume that it is ejected from the parent body with a velocity [FORMULA] in an arbitrary direction. The grain then follows a trajectory defined by gravitation and radiation forces. The ejection velocity depends upon the particle size. We take [FORMULA] (Sekanina & Larson 1984), which approximates the results of Probstein's (1969) two-phase dusty-gas dynamics for the acceleration by the expanding gas within tens of kilometers from the nucleus. The coefficient A and B depend upon many parameters such as the thermal velocity of the expanding gas. We used [FORMULA] s km-1 which is a good approximation of different values measured for the comets of the solar system (Sekanina & Larson 1984; Sekanina 1998). This gives an ejection velocity of [FORMULA] km s-1 for [FORMULA]m. The ejection velocity is smaller for larger grains which have smaller cross section area to mass ratio ([FORMULA] km s-1). We checked that any other realistic values for A and B give similar light curves within few percents.

2.2. Stellar parameter

The simulations have been performed with the mass, luminosity and radius of the central star set to solar values ([FORMULA], [FORMULA] and [FORMULA]). Simulation with other parameters could be possible. A larger mass for the central star would induce a shorter time scale; a larger luminosity would increase the effect of the radiation pressure on grains; a larger radius would decrease the relative extinction (Eq. 6). However, the conclusions do not strongly depend on these stellar properties. For instance, adopting the properties of an A5V star does not change the shape of the light curves. By scaling the production rate to the star luminosity, we found a quantitative change by less than a factor of two.

2.3. Grain properties

2.3.1. Extinction

The extinction cross-section of a dust grain is [FORMULA], where the extinction efficiency, [FORMULA] is slightly dependent on the particle size (s) and radiation wavelength ([FORMULA]). If [FORMULA] is the scattering efficiency (see Sect. 2.3.2) and [FORMULA] is absorption efficiency, [FORMULA]. We take [FORMULA] if [FORMULA], and [FORMULA] if [FORMULA], which is a good approximation for optical wavelengths and grains larger than 0.1 µm (Draine & Lee 1984).

The total extinction is hence calculated by adding the extinction due to all particles in the line of sight to the star. The optical depth [FORMULA] due to the dust is

[EQUATION]

where S is the projected area of the line of sight. In the simulation, one particle represents an optically thin cloud of several dust grains. The number of physical grains per particle in the simulation, [FORMULA], is

[EQUATION]

where [FORMULA] is the total dust mass, [FORMULA] is the dust density, and [FORMULA] is the number of particles in the simulation. [FORMULA] is set to a few [FORMULA] in order to keep reasonable computing time.

Because the cometary cloud is optically thick but its size is smaller than the size of the star, we mapped the stellar surface through a set of cells in polar coordinates. For each cell i, we calculate the optical depth [FORMULA] due to the particles within this cell of area [FORMULA] ([FORMULA]). The ratio of the flux observed through the cloud ([FORMULA]) to the initial stellar flux ([FORMULA]) is

[EQUATION]

The number of cells is the best compromise between the spatial resolution and the number of particles in the simulation. We take into account that the maximum contribution to [FORMULA] by each particle must be [FORMULA] in order to achieve an accurate result in spite of the quantization of the extinction. [FORMULA] and [FORMULA] are thus constrained by

[EQUATION]

The limb darkening is not taken into account in this work, because its effect is negligible.

2.3.2. Scattering

As already pointed out by Lamers et al. (1997), the main part of an occulting dust cloud is seen through a very small scattering angle. Thus, the total star light forward-scattered to the observer can be large because the phase function is strongly peaked to small angles for which the diffraction has the dominant contribution. The phase function for the diffraction is 1

[EQUATION]

where [FORMULA]. The phase function has been normalized by [FORMULA] and depends upon the wavelength [FORMULA]. Examples of such phase functions are shown in Lamers et al. (1997).

The scattered light ([FORMULA]) is evaluated by adding the contribution of each particle in the simulation.

[EQUATION]

[FORMULA] is the total extinction along the path from the star to the scattering grain and [FORMULA] is from the grain to the observer. The grain is at distance r from the star. For [FORMULA], the scattering efficiency [FORMULA] is assumed to be the diffraction efficiency for large grains: [FORMULA] (Pollack & Cuzzi, 1980). For small particles ([FORMULA]), we used the basic approximation [FORMULA] (van de Hulst, 1957). The result is very insensitive to this last assumption because forward scattering at small angle is largely dominated by diffraction on particles larger than the wavelength. Because each cloud represented by a particle is very thin, and the total extinction is small ([FORMULA], [FORMULA]), we assumed single scattering (no source function).

In fact, for very peaked forward-scattering function on very small scattering angle, the finite size of the star must be taken into account (especially when the dust cloud is seen superimposed on the star surface). Hence, we mapped the stellar surface by small arcs centered on the dust particle. We then calculate the total scattering by adding the contribution of each arc.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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