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

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3. Light curves

The light observed at a given time is the sum of two opposite effects: the increase of the brightness through the scattering by particles at small angle from the line of sight and the decrease of the brightness through the extinction (including absorption) by particles on the line of sight.

3.1. First order estimations

3.1.1. Extinction

From Eqs. 4 and 5, in the case [FORMULA] ([FORMULA]), and neglecting the inner extinction, we find an upper limit for the extinction:

[EQUATION]

For the particle size distribution given by Eq. 1 (Sect. 2.1), we have [FORMULA]. Adopting a dust production of [FORMULA] kg s-1 during 100 days, we find [FORMULA] kg. Hence [FORMULA]. This order of magnitude is close to the result of more elaborated calculations (Sect. 3.2). We can already conclude that this variation could be detected by a space photometric survey.

We note that the photometric variations of [FORMULA] Pic in 1981 was [FORMULA]. The above calculation gives an estimate of the lower limit of the needed production rate: [FORMULA] kg s-1. This is consistent with [FORMULA] kg s-1 obtained from the estimation of Lamers et al. (1997): [FORMULA]kg km-1 and with an assumed relative velocity between the dust and the nucleus of the comet [FORMULA] km s-1.

3.1.2. Scattering

If we take a comet at [FORMULA] AU from the star and an impact parameter on the line of sight of one half the stellar radius, then [FORMULA]. For the particle size distribution given by Eq. 1, the phase function is [FORMULA]. Therefore,

[EQUATION]

As a first conclusion, it is clear that both extinction and scattered light can be detected by very precise photometric survey, although scattering gives photometric variations an order of magnitude smaller than extinction. Extinction appears to be the major process observed when the comet is passing in front of the star. This event can thus be called an occultation.

3.2. Light curves

Taking into account the dust spatial distribution and the extinction within the tail, we calculated the light curves of a set of cometary occultations. From a given comet orbit, and dust production rate, we compute the full motion of each dust particle; the result is the variation of the star light as a function of time.

Two typical light curves resulting from the simulation are shown in Fig. 1 and 2. The majority of cometary occultations gives light curves with a very particular "rounded triangular" shape (Fig. 1). This occurs when the dense cometary head first occults the star and gives a very fast and sharp brightness decrease. It is then followed by the tail which gives an additional slow decrease. Similarly, the subsequent brightness increase is also sharp when the cometary head is going out of the occulting part. Then the increase slows down and the brightness returns to the normal level when the cometary tail is less and less dense in front of the star.

[FIGURE] Fig. 1. Plot of the photometric variations during a cometary occultation in red ([FORMULA] Å, thick solid line) and in blue ([FORMULA] Å, thick short-dashed line). The insert is a view from the top when the comet is crossing the line of sight ([FORMULA]) at the periastron. The production rate is [FORMULA] kg s-1 at 1 AU. The scattered light is given by the top thin lines, the extinction by the bottom thin lines (long-dashed line for [FORMULA] Å; dotted line for [FORMULA] Å). The total variation is plotted with the thick lines. This light curve presents the very specific "rounded triangular" shape observed in the majority of simulations of cometary occultations. The difference between the variation in the blue and in the red is negligible and less than 1%.

[FIGURE] Fig. 2. Same as Fig. 1. Here all the parameters are the same except that the longitude of the periastron is at 90o from the line of sight. This gives a production rate of [FORMULA] kg s-1 at 2 AU. The cometary tail is aligned with the line of sight; consequently the light curve presents a more symmetrical shape resembling a planetary occultation. The net color effect is hardly detectable and of the order of 1.5%.

However, in some configurations, the tail can be aligned with the line of sight. In these cases the light curves are more symmetric (Fig. 2). They can mimic planetary occultations (Fig. 3). Because of the noise, it will be difficult from such observations to differentiate between a comet and a planet.

[FIGURE] Fig. 3. Plot of the photometric variations due to a comet similar to the comet of the Fig. 2 (histogram). Here the result is given as a noisy observation with a photometric measurement every hour and an accuracy of [FORMULA]. For comparison, an occultation by a planet with a radius of 9 000 km and orbiting at 2 AU is plotted with limb darkening effect taken into account (thick line). The detected variation is very similar to a planetary occultation, the main difference being extended wings in the cometary case.

3.3. Color signatures

As seen in Sect. 2.3, the light variations show some color signature due to the optical properties of the grains. Particles with a size smaller than the wavelength are less efficient for extinction. Thus, extinction is smaller at larger wavelengths. The forward scattering is more peaked to the small angles at shorter wavelengths (Lamers et al. 1997). The scattering is thus larger at smaller wavelengths when the cometary cloud is occulting the star (a contrario , the scattering is larger at larger wavelength when the comet cloud is seen far from the star; but then the scattered light becomes negligible). As a result, there is a balance between the additional scattered light and the extinction; both are larger at smaller wavelengths. It is difficult to predict what will be the color effect on cometary occultation light curves.

In addition, most of the extinction and scattering due to the cometary dust is concentrated in the inner coma, where the number of large particles is larger because smaller particles are more efficiently ejected by radiation pressure (Sect. 2.1). As a consequence, in the optical, and with the size distribution assumed in Sect. 2.1, the light curves are barely dependent on the wavelength (or color band). The difference between variations in blue and in red is smaller than few percents. This color signature would be very difficult to observe. This is beyond the today technical feasibility.

However, for comets at small distances from the Sun ([FORMULA] AU), its appears that the dust size distribution is peaked at smaller sizes (Newburn & Spinrad 1985). This decrease of [FORMULA] with the heliocentric distance could be due to particle fragmentation. Although it is difficult to guess the size distribution for extra-solar comets, it is easy to understand that if small particles are more numerous (with [FORMULA]m), the occultation will show a larger color signature. We checked that a comet with a periastron at 0.3 AU and [FORMULA]m (Newburn & Spinrad 1985) gives larger extinction by about 15% in the blue than in the red (Fig. 4).

[FIGURE] Fig. 4. Same as Fig. 1. Here all the parameters are the same except that the periastron is at 0.3 AU and the particle size distribution is taken with [FORMULA]m. The production rate corresponds to [FORMULA] kg s-1 at 0.3 AU. Because there are more particles smaller than the wavelength, we see a dipper occultation in the blue by about 15%.

3.4. Conclusion

Photometric variations due to the stars' occultation by extra-solar comets could be detectable by photometric measurements with an accuracy of [FORMULA] - [FORMULA]. In most cases, the particular "rounded triangular" shape of the light curve can be an easy diagnostic for the presence of a comet. However, in some cases, the light curves can mimic an occultation by a more compact object like a classical extra-solar planet (Lecavelier des Etangs et al. 1997). The color signatures of cometary occultations can be too small to avoid the confusion. The detection of periodicity appears to be critical in the diagnostic of a planetary occultation. Alternatively, the polarization is another way to discriminate the two phenomena , because the light is scattered by the cometary dust roughly gathered in the same plane. But, with a level of at most few percents in polarization of solely the scattered light, this gives [FORMULA] 0.01% polarization in total and would also be difficult to detect. In cases of planetary and cometary occultations, spectroscopic follow-up observations should be planned to allow a better analysis of a suspected on-going detection.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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